Geometry Word Problems Practice Problems
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Practice
 If angles 1 and 2 are alternating angles, and the measure of angle 2 is 110°, what is the measure of angle 1?
 Angles F and G form a right angle. If angle F is 42°, what is the measure of angle G?
 Angles 1 and 2 are supplementary, and angles 1 and 3 are vertical. If the measure of angle 2 is 56°, what is the measure of angle 3?
 A triangle has one angle that is 13° and one angle that is 77°. What kind of triangle is it?
 If an isosceles triangle has one angle that is 100°, what are the measures of the other two angles?
 What is the base of a triangle whose area is 48 square units and whose height is 10 units?
 What is the perimeter of an isosceles triangle whose bases measure 12 inches and whose longest side is 20 inches?
 The angles of triangles JKL and MNO are identical. If the area of triangle JKL is four times the area of triangle MNO, and side JK is 16 meters long, how long is side MN?
 Right triangle XYZ has a hypotenuse of 10 centimeters. If the length of one leg is 6 centimeters, what is the length of the other leg?
 If the length of one leg of an isosceles triangle ABC is 8 feet, what is the length of the hypotenuse of triangle ABC?
 If a quadrilateral has four right angles, what type of quadrilateral could it be?
 What quadrilateral has perpendicular diagonals that are not congruent?
 One base of a trapezoid measures 16 meters, and the other measures 10 meters. If the height of the trapezoid is 4 meters, what is the area?
 If a rhombus has an area of 117 centimeters^{2} and a height of 9 centimeters, what is the base of the rhombus?
 The area of a square is 81 square feet. If the length of each side of the square is doubled, what is the area of the new square?
 What is the diameter of a circle whose radius is 100 inches?
 If the radius of a circle is 4 feet, what is the area of the circle?
 The circumference of a circle is 18π units. What is the diameter of the circle?
 If the area of a circle is 144π square units, what is the diameter of the circle?
 If the area of a circle is 361π centimeters^{2}, what is the circumference of the circle?
 If a rectangular prism has a length of 14 inches, a width of 9 inches, and a height of 10 inches, what is the volume of the prism?
 A sphere has a volume of 972π centimeters^{3}. What is the radius of the sphere?
 What is the volume of a cylinder whose radius is 8 feet and whose height is 12 feet?
 If a cone has a volume of 300π inches^{3} and a height of 36 inches, what is the radius of the cone?
 The area of one side of a cube is 2.56 meters^{2}. What is the volume of the cube?
 What is the surface area of a cylinder with a radius of 6 yards and a height of 18 yards?
 The surface area of a sphere is 213.16π millimeters^{3}. What is the radius of the sphere?
 If the volume of a cube is 343 inches^{3}, what is the surface area of the cube?
 A rectangular prism has a length of 3 meters and a width of 2.2 meters. If the prism has a volume of 27.06 meters^{3}, what is the surface area of the prism?
 The area of the base of a cylinder is 16π centimeters^{2}. If the height of the cylinder is 8 centimeters, what is the surface area of the cylinder?
Solutions
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We are told that the angles of triangles JKL and MNO are identical, which means that they are similar triangles. Draw and label triangles JKL and MNO:
We can use a proportion to find the length of side MN. Since the area of triangle JKL is four times the area of triangle MNO, the ratio of their sides . Cross multiply and divide: 4MN = 16, MN = 4 meters.
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Read the entire word problem.
We are given a radius.
Identify the question being asked.
We are looking for a diameter.
Underline the keywords and words that indicate formulas.
The words diameter and radius are keywords.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
The formula for diameter is d = 2r. Multiply the radius by 2 to find the diameter.
Write number sentences for each operation.
2(100)
Solve the number sentences and decide which answer is reasonable.
2(100) = 200 inches
Check your work.
Since diameter is twice the radius, divide the diameter by 2 to check that the radius is, in fact, 100 inches: = 2 inches.
 Read the entire word problem.

Read the entire word problem.
We are given a circumference.
Identify the question being asked.
We are looking for a diameter.
Underline the keywords and words that indicate formulas.
The words circumference and diameter are keywords.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
The formula for circumference is C = πd, so we can find the diameter by dividing the circumference by π.
Write number sentences for each operation.
Solve the number sentences and decide which answer is reasonable.
= 18 units
Check your work.
Multiply the diameter by π. This should give us back the circumference: (18)(π) = 18π, so our answer is correct.
 Read the entire word problem.
We are given an area.
Identify the question being asked.
We are looking for a diameter.
Underline the keywords and words that indicate formulas.
The words area and diameter are keywords.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
The formula for the area of a circle is A = πr^{2}, so we can find the radius by dividing the area by π, and then taking the square root of the quotient. Once we have the radius, we can multiply it by 2 to find the diameter.
Write number sentences for each operation.
Divide the area by π:
= 18 units
Solve the number sentences and decide which answer is reasonable.
= 144 square units
Write number sentences for each operation.
Find the radius by taking the square root of 144:
√144
Solve the number sentences and decide which answer is reasonable.
√144 = 12
The radius of the circle is 12 units. To find the diameter, multiply the radius by 2.
Write number sentences for each operation.
(12)(2)
Solve the number sentences and decide which answer is reasonable.
(12)(2) = 24 units
Check your work.
Divide the diameter by 2 to find the radius. Then, square the radius and multiply by π. This should give us back the area, 144π square units. = 12, π(12)^{2} = 144π square units.
 Read the entire word problem.
We are given an area.
Identify the question being asked.
We are looking for a circumference.
Underline the keywords and words that indicate formulas.
The words area and circumference are keywords.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
The formula for the area of a circle is A = πr^{2}, so we can find the radius by dividing the area by π, and then taking the square root of the quotient. Once we have the radius, we can multiply it by 2π to find the circumference.
Write number sentences for each operation.
Divide the area by π:
Solve the number sentences and decide which answer is reasonable.
= 361 square unitsWrite number sentences for each operation.
Find the radius by taking the square root of 361:
√361
Solve the number sentences and decide which answer is reasonable.
√361 = 19
The radius of the circle is 19 units. To find the circumference, multiply the radius by 2π.
Write number sentences for each operation.
(19)(2π)
Solve the number sentences and decide which answer is reasonable.
(19)(2π) = 38πunits
Check your work.
Divide the circumference by 2π to find the radius. Then, square the radius and multiply by π. This should give us back the area, 361π square units. = 19, π(19)^{2} = 361π square units.
 Read the entire word problem.
We are given the length, width, and height of a rectangular prism.
Identify the question being asked.
We are looking for its volume.
Underline the keywords and words that indicate formulas.
The words rectangular prism and volume are keywords. We will use the formula for the volume of a rectangular prism.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
The formula for the volume of a rectangular prism is V = lwh, so we can multiply the three given values to find the volume.
Write number sentences for each operation.
(14)(9)(10)
Solve the number sentences and decide which answer is reasonable.
(14)(9)(10) = 1,260 inches^{3}
Check your work.
Divide the volume by the length and the width. This should give us back the height: = 10 inches.
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We are given the volume of a sphere.
Identify the question being asked.
We are looking for its radius.
Underline the keywords and words that indicate formulas.
The words sphere and volume are keywords. We will use the formula for the volume of a sphere prism to find the radius of the sphere.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
The formula for the volume of a sphere is V = πr^{3}. Divide both sides of the equation by π and take the cube root of both sides: r = 3. First, multiply the volume by 3. Then, divide that product by 4π. Finally, take the cube root of the result.
Write number sentences for each operation.
(3)(972π)
Solve the number sentences and decide which answer is reasonable.
(3)(972π) = 2,916π centimeters^{3}
Write number sentences for each operation.
Divide 2,916 by 4π:
Solve the number sentences and decide which answer is reasonable.
= 728
Write number sentences for each operation.
Take the cube root of 729:
3√729
Solve the number sentences and decide which answer is reasonable.
3√729 = 9 centimeters
The radius of the sphere is 9 centimeters.
Check your work.
Plug the radius into the formula for the volume of a sphere. πr^{3} = π(9)^{3} = (729)π = 972π centimeters^{3}.
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We are given the radius and height of a cylinder.
Identify the question being asked.
We are looking for its volume.
Underline the keywords and words that indicate formulas.
The words cylinder and volume are keywords. We will use the formula for the volume of a cylinder.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
The formula for the volume of a cylinder is V = πr^{2}h, so we can square the radius and multiply by the height to find the volume.
Write number sentences for each operation.
π(8)^{2}(12)
Solve the number sentences and decide which answer is reasonable.
π(8)^{2}(12) = 768π feet^{3}
Check your work.
Divide the volume by the height and π. Then, take the square root, and we should have the radius: = 64, √64 = 8 feet.
 Read the entire word problem.
We are given the volume and height of a cone.
Identify the question being asked.
We are looking for its radius.
Underline the keywords and words that indicate formulas.
The words cone and volume are keywords. We will use the formula for the volume of a cone.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
The formula for the volume of a cone is V = πr^{2}h, so we can rewrite this equation to solve for r, the radius. Multiply both sides of the equation by 3 and divide by π and h. Then, take the square root of both sides.
Write number sentences for each operation.
First, multiply the volume by 3:
(300π)(3)
Solve the number sentences and decide which answer is reasonable.
(300π)(3) = 900π inches^{3}
Write number sentences for each operation.
Next, divide that product by π and h:
Solve the number sentences and decide which answer is reasonable.
= 25 inches^{2}
Write number sentences for each operation.
Finally, take the square root:
√25
Solve the number sentences and decide which answer is reasonable.
√25 = 5 inches
Check your work.
Plug the radius and height into the formula for volume of a cone. πr^{2}h = π(5)^{2}(36) = (25)π(36) = (25π)(12) = 300π inches^{3}.
 Read the entire word problem.
We are given the area of one side of a cube.
Identify the question being asked.
We are looking for the volume of the cube.
Underline the keywords and words that indicate formulas.
The words area, cube, and volume are keywords. We will use the formula for the area of a square and the formula for the volume of a cone.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
We must use the area of one side of the cube to find the length of one edge of the cube. Then, we can use that length to find the volume of the cube. Since the formula for the area of a square is A = s^{2}, take the square root of the area to find the length of one side, or edge. The formula for the volume of a cube is V = e^{3}, so we must then take the length of one edge and cube it to find the volume.
Write number sentences for each operation.
First, take the square root of the area:
√2.56
Solve the number sentences and decide which answer is reasonable.
√2.56 = 1.6 meters
Write number sentences for each operation.
Now, cube the length of one edge:
(1.6)^{3}
Solve the number sentences and decide which answer is reasonable.
(1.6)^{3} = 4.096 meters^{3}
Check your work.
Take the cube root of the volume to find the length of one edge, and then square that to find the area of one side of the cube: 3√4.096 = 1.6, (1.6) ^{2}= 2.56 meters^{2}.
 Read the entire word problem.
We are given the radius and height of a cylinder.
Identify the question being asked.
We are looking for its surface area.
Underline the keywords and words that indicate formulas,.
The words cylinder and surface area are keywords. We will use the formula for the surface area of a cylinder.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
The formula for the surface area of a cylinder is SA = 2πr^{2} + 2πrh, so we must begin by squaring the radius and multiplying it by 2π. Then, we must multiply the radius and the height by 2π. Finally, we must add those two products together.
Write number sentences for each operation.
First, square the radius and multiply it by 2π:
[(6)^{2}](2π)
Solve the number sentences and decide which answer is reasonable.
[(6)^{2}](2π) = (36)(2π) = 72π
Write number sentences for each operation.
Now, multiply the radius and the height by 2π:
(6)(18)(2π)
Solve the number sentences and decide which answer is reasonable.
(6)(18)(2π) = 216π
Write number sentences for each operation.
Add the two products:
72π + 216π
Solve the number sentences and decide which answer is reasonable.
72π + 216π = 288π
Check your work.
Plug both the radius and the height into the surface area formula and do the calculations again to insure your answer is correct: 2π(6)^{2} + 2π(6)(18) = 72π + 216π = 288π.
 Read the entire word problem.
We are given the surface area of a sphere.
Identify the question being asked.
We are looking for its radius.
Underline the keywords and words that indicate formulas.
The words sphere and surface area are keywords. We will use the formula for the surface area of a sphere to find the radius of the sphere.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
The formula for the surface area of a sphere is SA = 4πr^{2}, so we can rewrite it to solve for the radius by dividing both sides by 4π, and then taking the square root of both sides.
Write number sentences for each operation.
First, divide the surface area by 4π:
Solve the number sentences and decide which answer is reasonable.
= 53.29
Write number sentences for each operation.
Now, take the square root of 53.29:
√53.29
Solve the number sentences and decide which answer is reasonable.
√53.29 = 7.3
Check your work.
Plug the radius into the surface area formula to find the surface area of the sphere: 4πr^{2} = 4π(7.3)^{2} = 4π(53.29) = 213.16π millimeters^{3}.
 Read the entire word problem.
We are given the volume of a cube.
Identify the question being asked.
We are looking for its surface area.
Underline the keywords and words that indicate formulas.
The words cube, volume, and surface area are keywords. We will use the formula for the volume of a cube to find the edge of the cube, and then use the formula for surface area of a cube.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
The formula for the volume of a cube is V = e^{3}, so we can find the edge of the cube by taking the cube root of the volume. Since the formula for surface area of a cube is SA = 6s^{2}, once we have the length of one edge, or side, we can square it and multiply by 6 to find the surface area.
Write number sentences for each operation.
First, take the cube root of 343:
3√343
Solve the number sentences and decide which answer is reasonable.
√343 = 7 inches
Write number sentences for each operation.
Now, square that length and multiply by 6:
6(7)^{2}
Solve the number sentences and decide which answer is reasonable.
6(7)^{2} = 6(49) = 294 inches^{2}
Check your work.
Divide the surface area by 6 and take the square root. This will give us the length of one edge of the cube. Cube that length to get back the volume of the cube: = 49, 49^{3} = 343 inches3.
 Read the entire word problem.
We are given the length, width, and volume of a rectangular prism.
Identify the question being asked.
We are looking for its surface area.
Underline the keywords and words that indicate formulas.
The words rectangular prism, volume, and surface area are keywords. We will use the formula for the volume of a rectangular prism to find the height of the prism, and then use the formula for the surface area of a rectangular prism.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
The formula for the volume of a rectangular prism is V = lwh. We can rewrite that as h = by dividing both sides of the equation by lw. Once we have the height of the rectangular prism, we can plug it, the length, and the width into the formula for the surface area of a rectangular prism: SA = 2(lw +wh + lh).
Write number sentences for each operation.
First, divide the volume by the product of the length and width to find the height:
Solve the number sentences and decide which answer is reasonable.
= 4.1 meters
rite number sentences for each operation.
The height of the prism is 4.1 meters. Now, plug the values of the length, width, and height into the formula for the surface area of a rectangular prism:
2[(3)(2.2) + (2.2)(4.1) + (3)(4.1)]
Solve the number sentences and decide which answer is reasonable.
2[(3)(2.2) + (2.2)(4.1) + (3)(4.1)] = 2(6.6 + 9.02 + 12.3) = 2(27.92) = 55.84 meters^{2}
Check your work.
Check that the height you found, 4.1 meters, is correct by multiplying it by the length and width to see if it equals the given volume, 27.06 meters^{3}: (3)(2.2)(4.1) = 27.06 meters^{3}.
 Read the entire word problem.
We are given the height and the area of the base of a cylinder.
Identify the question being asked.
We are looking for its surface area.
Underline the keywords and words that indicate formulas.
The words cylinder, area, and surface area are keywords. We will use the formula for the surface area of a cylinder.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
The formula for the surface area of a cylinder is SA = 2πr^{2} + 2πrh. The bases of a cylinder are circles, so the first part of the surface area formula, 2πr^{2}, is equal to twice the area of the base. We are given the area of the base, so we can multiply that by 2 to find the first part of the surface area of the cylinder. To find the radius of the cylinder, we must divide the area of one base by π and then take the square root. Once we have the radius, we can multiply it by 2π and by the height, and this will give us the second part of the surface area. To find the total surface area, we will add the two parts together.
Write number sentences for each operation.
First, multiply the area of one base by 2:
(16π)(2)
Solve the number sentences and decide which answer is reasonable.
(16π)(2) = 32π
Write number sentences for each operation.
We have the first part of the surface area. Next, find the radius of the cylinder. Divide the area of one base by π and then take the square root:
Solve the number sentences and decide which answer is reasonable.
= 16
Write number sentences for each operation.
√16
Solve the number sentences and decide which answer is reasonable.
√16 = 4 centimeters
Write number sentences for each operation.
The radius is 4 centimeters. Multiply it by 2π and by the height to find, the second part of the surface area.
(2π)(4)(8)
Solve the number sentences and decide which answer is reasonable.
(2π)(4)(8) = 64π
Write number sentences for each operation.
Finally, add the first part of the surface area, the area of the bases, to the second part of the surface area, the area of the curved surface:
32π + 64π
Solve the number sentences and decide which answer is reasonable.
32π + 64π = 96π centimeters^{2}
Check your work.
Plug both the radius and the height into the surface area formula:
2π(4)^{2} + 2π(4)(8) = 32π + 64π = 96π centimeters^{2}.
We are given the measure of one of two alternating angles.
Identify the question being asked.
We are looking for the measure of the other alternating angle.
Underline the keywords and words that indicate formulas.
The word alternating tells us that angles 1 and 2 are congruent—their measures are the same.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
We don't need to perform any operations or write any number sentences. Alternating angles are congruent, so since angle 2 is 110°, angle 1 is 110°.
Check your work.
We didn't perform any operations, so we don't have any work to check. Alternating angles are also known as corresponding angles, and the definition of corresponding angles is two or more congruent angles that have similar positions in a shape or figure. Our answer is correct.
We are given the measure of angle F and told that it forms a right angle with angle G.
Identify the question being asked.
We are looking for the measure of angle G.
Underline the keywords and words that indicate formulas.
The phrase right angle tells us that angles F and G are complementary—their measures sum to 90°.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
Since angles F and G are complementary, we must subtract the measure of angle F from 90° to find the measure of angle G.
Write number sentences for each operation.
90 – 42
Solve the number sentences and decide which answer is reasonable.
90 – 42 = 48°
Check your work.
There are 90° in a right angle, so the measures of angles F and G should add up to 90: 42 + 48 = 90°.
We are given the measure of angle 2 and told that angles 1 and 2 are supplementary. We are also told that angles 1 and 3 are vertical.
Identify the question being asked.
We are looking for the measure of angle 3.
Underline the keywords and words that indicate formulas.
The word supplementary tells us that the measures of angles 1 and 2 add up to 180°, and the word vertical tells us that angles 1 and 3 are congruent.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
Since angles 1 and 2 are supplementary, we must subtract the measure of angle 2 from 180° to find the measure of angle 1.
Write number sentences for each operation.
180 – 56
Solve the number sentences and decide which answer is reasonable.
180 – 56 = 124°
Angle 1 is 124°. Angles 1 and 3 are vertical angles, so angle 3 is also 124°.
Check your work.
We can check that angle 1 is 124° by adding its measure to the measure of angle 2, since these angles are supplementary: 124 + 56 = 180°. Vertical angles are defined as congruent angles that are formed by the intersection of two lines, so angle 3 does, in fact, equal angle 1.
We are given the measures of two angles of a triangle.
Identify the question being asked.
We are looking for the name of this kind of triangle.
Underline the keywords and words that indicate formulas.
There are no keywords in this problem.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
The two angle measures are not 60° and they are not equal, so this triangle is not equilateral or isosceles. We must add the two angles together and subtract from 180° to find the measure of the third angle. Then, we will be able to determine if the triangle is acute, obtuse, or right.
Write number sentences for each operation.
First, add the two given angles:
13 + 77
Solve the number sentences and decide which answer is reasonable.
13 + 77 = 90°
Subtract this sum from 180°, the number of degrees in a triangle, to find the measure of the third angle.
Write number sentences for each operation.
180 – 90
Solve the number sentences and decide which answer is reasonable.
180 – 90 = 90°
The third angle of the triangle is a right angle, which means that this is a right triangle.
Check your work.
We can check that the three angles add to 180°; 13 + 77 + 90 = 180°, so the third angle is, in fact, a right angle, and this triangle is a right triangle.
We are given the measure of one angle of an isosceles triangle.
Identify the question being asked.
We are looking for the measures of the other two angles.
Underline the keywords and words that indicate formulas.
The keyword isosceles tells us that two angles in the triangle are congruent.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
The given angle is 100°. It cannot be one of the base angles of the triangle, because there cannot be two obtuse angles in a triangle (since there are only 180° in a triangle). The other two angles must be the base angles. Subtract the measure of the given angle from 180° to find the total measure of the other two angles. Then, divide that difference by 2 to find the measure of each of those angles.
Write number sentences for each operation.
First, subtract 100 from 180:
180 – 100
Solve the number sentences and decide which answer is reasonable.
180 – 100 = 80°
The two base angles combined equal 80°. Divide 80 by 2 to find the measure of each base angle.
Write number sentences for each operation.
Solve the number sentences and decide which answer is reasonable.
The other two angles each measure 40°.
Check your work.
We can check that the three angles add up to 180°: 100 + 40 + 40 = 180°. The three angles total 180°, and two of the angles are the same in measure, so our answer is correct.
We are given the area and the height of a triangle.
Identify the question being asked.
We are looking for its base.
Underline the keywords and words that indicate formulas.
The keywords base, area, height, and triangle tell us that we need to use the formula for area of a triangle.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
We are given the area and the height. We can rewrite the formula A = by dividing both sides of . The base, b, is equal to .
Write number sentences for each operation.
First, multiply the area by 2:
2(48)
Solve the number sentences and decide which answer is reasonable.
2(48) = 96 square units
Write number sentences for each operation.
Now, divide that by the height:
Solve the number sentences and decide which answer is reasonable.
The base of the triangle is 9.6 units.
Check your work.
Since A = , check that half the product of the base and the height equals the area, 48 square units: square units.
We are given the measures of the bases and the longest side of an isosceles triangle.
Identify the question being asked.
We are looking for the perimeter of the triangle.
Underline the keywords and words that indicate formulas.
We are asked to find the perimeter, and the keywords isosceles triangle tell us that the base angles of the triangle are equal. We have been given all three sides of the triangle, and we must add them to find the perimeter.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
The perimeter of a triangle is equal to the sum of the lengths of the sides, so we must add.
Write number sentences for each operation.
14 + 14 + 20
Solve the number sentences and decide which answer is reasonable.
14 + 14 + 20 = 44 inches
The perimeter of the triangle is 44 inches.
Check your work.
Since the triangle is isosceles, if we subtract base lengths from the perimeter, we should be left with the other side of the triangle, which is 20 inches long: 44 – 12 – 12 = 20 inches.
We are given the measures of one leg and the hypotenuse of a right triangle.
Identify the question being asked.
We are looking for the measure of the other leg.
Underline the keywords and words that indicate formulas.
We are told we are working with a right triangle, which the word hypotenuse also indicates.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
Since we have a right triangle and the lengths of two sides, we can use the Pythagorean theorem to find the length of the missing side. Rewrite the formula a^{2} + b^{2} = c^{2} as b^{2} = c^{2} – a^{2}, since we are looking for the length of a leg.
Write number sentences for each operation.
First, find the difference between the square of the hypotenuse and the square of one leg:
(10)^{2} – (6)^{2}
Solve the number sentences and decide which answer is reasonable.
(10)^{2} – (6)^{2} = 100 – 36 = 64
Write number sentences for each operation.
The square of the leg is 64, so to find the length of the leg, we must take the square root of 64: √64
Solve the number sentences and decide which answer is reasonable.
√64 = 8
The length of the other leg is 8 centimeters.
Check your work.
The square of the hypotenuse should equal the sum of the squares of the legs: (6)^{2} + (8)^{2} = (100)^{2}, 36 + 64 = 100, 100 = 100.
We are given the measures of one leg of a triangle.
Identify the question being asked.
We are looking for the measure of the hypotenuse.
Underline the keywords and words that indicate formulas.
The word hypotenuse indicates we are working with a right triangle.
Since the triangle is also isosceles, it must be an isosceles right triangle.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
Since the triangle is isosceles, both legs are 8 feet. Use the formula a^{2} + b^{2} = c^{2} to find c, the length of the hypotenuse. We will need to square the legs and add those squares. Then, we will need to take the square root of that sum.
Write number sentences for each operation.
First, square the legs and add them:
(8)^{2} + (8)^{2}
Solve the number sentences and decide which answer is reasonable.
(8)^{2} + (8)^{2} = 64 + 64 = 128
Write number sentences for each operation.
The square of the hypotenuse is 128, so to find the length of the hypotenuse, we must take the square root of 128:
√128
Solve the number sentences and decide which answer is reasonable.
√128 = 8√2 feet.
The hypotenuse of triangle ABC is 8√2 feet.
Check your work.
The square of the hypotenuse should equal the sum of the squares of the legs: (8)^{2} + (8)^{2} = (8√2)^{2}, 64 + 64 = 128.
We are told that a quadrilateral has four right angles.
Identify the question being asked.
We are looking for the type of quadrilateral that it could be.
Underline the keywords and words that indicate formulas.
The keywords four right angles tell us that this quadrilateral is either a rectangle or a square.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
There are no operations to perform. A rectangle has four right angles, and a square is a type of rectangle, so this quadrilateral could be a square or a rectangle.
Check your work.
The definition of a rectangle is a parallelogram with four congruent, 90° angles and congruent diagonals, and a square is a rectangle, so this answer is correct.
We are told that a quadrilateral has perpendicular diagonals that are not congruent.
Identify the question being asked.
We are looking for the type of quadrilateral that it could be.
Underline the keywords and words that indicate formulas.
The keywords perpendicular diagonals tell us that this quadrilateral is a rhombus or a square, and the keywords not congruent tell us that it is not a square, since the diagonals of a square are congruent.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
There are no operations to perform. A rhombus has perpendicular diagonals, as does a square since a square is a type of rhombus. But a square has congruent diagonals, so this quadrilateral must be a rhombus.
Check your work.
The definition of a rhombus is a parallelogram with four congruent sides and perpendicular diagonals that bisect their angles, while a square has all of the properties of a rhombus and a rectangle, and the diagonals of a rectangle are congruent.
We are given the measures of the bases and height of a trapezoid.
Identify the question being asked.
We are looking for the area of the trapezoid.
Underline the keywords and words that indicate formulas.
The words area and trapezoid indicate that we must use the formula for area of a trapezoid: .
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
First, we must add the bases and multiply by onehalf. Then, we must multiply that product by the height.
Write number sentences for each operation.
(16 + 10)
Solve the number sentences and decide which answer is reasonable.
(16 + 10) = (26) = 13 meters
Write number sentences for each operation.
Now multiply that product by the height, 4:
(13)(4)
Solve the number sentences and decide which answer is reasonable.
(13)(4) = 52 meters
The area of the trapezoid is 52 meters.
Check your work.
Divide the area by half the sum of the baseotient should equal the height, 4 meters: 4 meters.
We are given the area and height of a rhombus.
Identify the question being asked.
We are looking for its base.
Underline the keywords and words that indicate formulas.
The words area and rhombus indicate that we must use the formula for area of a parallelogram, since a rhombus is a parallelogram: A = bh.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
Rewrite the formula to solve for b, the base of the rhombus: b = .
Divide the area by the height.
Write number sentences for each operation.
Solve the number sentences and decide which answer is reasonable.
= 13 centimeters
The base of the rhombus is 13 centimeters.
Check your work.
Since the area of a rhombus is equal to the product of its base and its height, multiply 13 by 9 to see if it equals the area of the rhombus, 117 centimeters^{2}: (13)(9) = 117 centimeters^{2}.
We are given the area of a square and how it changes.
Identify the question being asked.
We are looking for the new area of the square.
Underline the keywords and words that indicate formulas.
The words area and square indicate that we must use the formula for area of a square, to find the length of one side of the square: A = s^{2}.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
We can take the square root of the area to find the length of one side of the square. Then, we can double that length and use the area formula again to find the area of the new square.
Write number sentences for each operation.
First, take the square root of the area:
√81
Solve the number sentences and decide which answer is reasonable.
√81 = 9 feet
Write number sentences for each operation.
One side of the original square is 9 feet. To find the length of one side of the new square, multiply 9 by 2:
(9)(2)
Solve the number sentences and decide which answer is reasonable.
(9)(2) = 18 feet
Write number sentences for each operation.
One side of the new square is 18 feet. To find the area of the new square, square 18:
18^{2}
Solve the number sentences and decide which answer is reasonable.
18^{2} = 324 square feet
Check your work.
If the length of a side of a square doubles, the area of the square increases by four times. Multiply the original area, 81 square feet, by 4 to see if it equals the new area: (81)(4) = 324 square feet.
We are given a radius.
Identify the question being asked.
We are looking for an area.
Underline the keywords and words that indicate formulas.
The words radius and area are keywords.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
The formula for the area of a circle is A = 2πr^{2}. Square the radius and multiply by π to find the area.
Write number sentences for each operation.
π(4)^{2}
Solve the number sentences and decide which answer is reasonable.
π(4)^{2} = 16π inches^{2}
Check your work.
Divide the area by π, and then take the square root of the quotient. This should give us back the radius, 4 feet: = 16, √16 = 4 feet.

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