Example
If angles A and B are vertical angles, and the measure of angle A is 65°, what is the measure of angle B?
Read the entire word problem.
We are given the measure of angle A, and we are told that it and angle B are vertical angles.
Identify the question being asked.
We are looking for the measure of angle B.
Underline the keywords and words that indicate formulas.
The word vertical tells us that angles A and B 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. Vertical angles are congruent, so since angle A is 65°, angle B is 65°.
Check your work.
We didn't perform any operations, so we don't have any work to check. We can check the definition of vertical angles, which states that two vertical angles are equal in measure.
Example
If angles A and B are complementary angles, and the measure of angle A is 12°, what is the measure of angle B?
Read the entire word problem.
We are given the measure of angle A, and we are told that it and angle B are complementary angles.
Identify the question being asked.
We are looking for the measure of angle B.
Underline the keywords and words that indicate formulas.
The word complementary tells us that the measures of angles A and B total 90°.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
Since complementary angles add to 90° and we have the measure of one angle, we must use subtraction to find the measure of the other angle.
Write number sentences for each operation.
90 – 12
Solve the number sentences and decide which answer is reasonable.
90 – 12 = 78°
Check your work.
Since complementary angles add to 90°, the sum of the measures of angles A and B should be 90°: 78 + 12 = 90°.
Not all geometry questions require computations. Sometimes, a word problem may test our knowledge of a rule or property. Look closely at the question being asked to see if it is asking you for a number, word, or phrase.
Example
If a triangle contains a 91° angle, what kind of triangle is it?
Read the entire word problem.
We are given the measure of one angle of a triangle.
Identify the question being asked.
We must identify what kind of triangle it is.
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.
A 91° angle is an obtuse angle, because it is greater than 90° in measure. This means that the triangle is an obtuse triangle. We don't need to write any number sentences.
Check your work.
The definition of an obtuse triangle is a triangle that contains an angle that is greater than 90°, so our answer is correct.
Caution
Even if a word problem does require addition, subtraction, multiplication, or division, sometimes our answer can still be a word or phrase and not a number. The first step of the wordproblemsolving process is always the most important: read the entire word problem. Equally important is determining the question being asked so that we know how to go about solving the problem.

Example
If a triangle contains two 33° angles, in what two ways can this triangle be classified?
Read the entire word problem.
We are given the measure of two angles of a triangle.
Identify the question being asked.
We are looking for two ways to name the 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.
Triangles can be classified as acute, right, obtuse, equilateral, and isosceles. A triangle is equilateral if all three angles measure 60°. Since two of the angles in this triangle are 33°, the triangle is not equilateral. However, the triangle is isosceles, because an isosceles triangle has two congruent angles. We have one name for this triangle, but we need another. In order to tell if this triangle is acute, right, or obtuse, we must find the measure of the third angle. There are 180° in a triangle, so we must find the sum of the two 33° angles, and subtract it from 180°.
Write number sentences for each operation.
33 + 33
Solve the number sentences and decide which answer is reasonable.
33 + 33 = 66°
Write number sentences for each operation.
Now subtract that sum from 180°:
180 – 66
Solve the number sentences and decide which answer is reasonable.
180 – 66 = 114°
Since this angle is greater than 90°, the triangle is also an obtuse triangle.
Check your work.
There are 180° in a triangle, so we can check that the third angle is 114° by finding the sum of the measures of the three angles: 33 + 33 + 114 = 180°.
Triangles: Subject Review
The formula for area of a triangle is A = bh, where A is the area, b is the base of the triangle, and h is the height of the triangle. If the base of a triangle is 4 feet and the height is 5 feet, then the area of the triangle is equal to (4)(5) = (20) = 10 feet^{2}.
The perimeter of a triangle is equal to the sum of the lengths of its three sides. If a triangle is equilateral, then its perimeter can be found by the formula P = 3s, where s is the length of one side of the triangle. If a triangle is isosceles, then its perimeter is equal to twice the length of one of the sides that are equal plus the length of the remaining side. A triangle with sides that measure 4 inches, 8 inches, and 6 inches has a perimeter of 4 + 8 + 6 = 18 inches.
If two triangles are similar, we can use the ratio of their sides or the ratio of their areas to find any missing measurements. If we know that the ratio of triangle A to triangle B is 2:1 and that the area of triangle A is 24 square units, then the area of triangle B, x, can be found by solving this proportion: , 2x = 24, x = 12. The area of triangle B is 12 square units.
If we are given the lengths of two sides of a right triangle, we can find the length of the missing side using the Pythagorean theorem: a^{2} + b^{2} = c^{2}, where a and b are the bases of the right triangle and c is the hypotenuse. If one base of a right triangle measures 3 feet and the other measures 4 feet, then (3)^{2} + (4)^{2} = c^{2}, 9 + 16 = c^{2}, 25 = c^{2}, and c = 5. The hypotenuse of the triangle is 5 feet.
Fuel for Thought
Area is the amount of twodimensional space that a surface covers. Area is always expressed in square units.
Perimeter is the distance around an area. Perimeter is always expressed in linear units.
Similar triangles have identical angles. All three corresponding angles are congruent, although the sides of the triangles may not be congruent.
The Pythagorean theorem states that the sum of the squares of the bases of a right triangle is equal to the square of the hypotenuse of the right triangle: a^{2} + b^{2} = c^{2}.

Most area word problems will contain the word area. Sometimes these problems give the base and height of a shape, and sometimes, they give the area and one of the two measurements and ask you for the other measurement. Perimeter word problems often contain the word perimeter, but a perimeter word problem may also ask you to find the distance around a figure.
Example
What is the area of a triangle whose base is 3 meters and whose height is 22 meters?
Read the entire word problem.
We are given the base and height of a triangle.
Identify the question being asked.
We are looking for the area of the triangle.
Underline the keywords and words that indicate formulas.
The phrase area of a triangle tells us that we must use the area formula for triangles.
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 base and the height. The area of a triangle is the product of , the base, and the height, so we must multiply.
Write number sentences for each operation.
(3)(22)
Solve the number sentences and decide which answer is reasonable.
(3)(22) = (66) = 33 square meters
Check your work.
Since we used multiplication to find our answer, we can use division to check it. Divide the area by the base, and that value should equal half the height: = 11, which is half of 22.
Example
If an equilateral triangle has a perimeter of 111 inches, what is the length of one side of the triangle?
Read the entire word problem.
We are given the perimeter of an equilateral triangle.
Identify the question being asked.
We are looking for the length of one side of the triangle.
Underline the keywords and words that indicate formulas.
The word perimeter tells us that we have the distance around a triangle, and the phrase equilateral triangle tells us that every side of the triangle is equal in measure. These words together mean that we can use the formula for perimeter of an equilateral triangle, P = 3s.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
Rewrite the formula for perimeter of an equilateral triangle to solve for s, the side of the triangle: s =
Write number sentences for each operation.
Divide the perimeter by 3:
Solve the number sentences and decide which answer is reasonable.
= 37 inches
Check your work.
Since we used division to find our answer, we can use multiplication to check it. The perimeter of the equilateral triangle should be equal to three times the length of one side: (3)(37) = 111 inches.
Almost all word problems that involve similar triangles will include the word similar, but remember that similar triangles are two triangles whose angles are identical. If a word problem describes two triangles as having identical angles, be prepared to set up a ratio to find a missing side or area of one of the triangles. If a word problem is about similar triangles, draw a picture to help you match up corresponding sides.
Example
Triangles ABC and DEF are similar. If the length of side AB is 12 feet, the length of side DE is 30 feet, and the length of BC is 8 feet, what is the length of side EF?
Since this is a similar triangle word problem, we'll solve it by drawing a picture. Start by drawing two triangles. It's not important how they look, but it is important how we label them. We must be sure to match angle A with angle D, angle B with angle E, and angle C with angle F:
Next, label AB 12 feet, DE 30 feet, and BC 8 feet:
We can see now that sides AB and DE are corresponding and sides BC and EF are corresponding. Since the triangles are similar, the ratio of side AB to side DE is equal to the ratio of side BC to side EF: . Cross multiply and divide: 12EF = 240, EF = 20 feet.
We can identify a problem that involves a right triangle if we see the word hypotenuse. The word problem may also state that the triangle is a right triangle, or it may simply state that the triangle contains a 90° or right angle. If a problem asks for the missing side of a right triangle, you will likely need to use the Pythagorean theorem.
Example
The legs of triangle GHI form a right angle. If the length of one leg is 15 feet and the length of the other leg is 20 feet, what is the length of the hypotenuse?
Read the entire word problem.
We are given the lengths of the legs of a triangle that has a right angle.
Identify the question being asked.
We are looking for the length of the hypotenuse.
Underline the keywords and words that indicate formulas.
This problem contains many keywords. The legs of the triangle form a right angle, which means that this is a right triangle. The word hypotenuse also tells us that this is a right triangle, and in order to find the length of the hypotenuse, we will have to use the Pythagorean theorem: a^{2} + b^{2} = c^{2}.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
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:
(15)^{2} + (20)^{2}
Solve the number sentences and decide which answer is reasonable.
(15)^{2} + (20)^{2} = 225 + 400 = 625
Write number sentences for each operation.
The answer 625 is the square of the hypotenuse, so to find the length of the hypotenuse, we must take the square root of 625:
√625
Solve the number sentences and decide which answer is reasonable.
√625 = 25 feet
The hypotenuse of triangle GHI is 25 feet.
Check your work.
The square of the hypotenuse should equal the sum of the squares of the legs: (15)^{2} + (20)^{2} = (25)^{2}, 225 + 400 = 625, 625 = 625.
Inside Track
Only a right triangle has a hypotenuse, so even if you are not told that a triangle is right, if you are asked to find the length of the hypotenuse, you are working with a right triangle, and you will need to use the Pythagorean theorem.

Polygons and Quadrilaterals: Subject Review
The sum of the interior angles of a polygon is equal to 180(s – 2), where s is the number of sides of the polygon. If a polygon has ten sides, then the sum of its interior angles is 180(10 – 2) = 180(8) = 1,440°.
There are six major types of quadrilaterals: parallelograms, rhombi, rectangles, squares, trapezoids, and isosceles trapezoids. By analyzing the properties of a quadrilateral, we can determine if the quadrilateral is a parallelogram, rhombus, rectangle, square, trapezoid, isosceles triangle—or just a quadrilateral.
Fuel for Thought
An interior angle is an angle found within a closed figure.
A polygon is a closed figure made up of line segments.
A quadrilateral is a polygon with four sides.
A parallelogram is a quadrilateral whose opposite sides are parallel and congruent. Its opposite angles are also congruent, consecutive angles are supplementary, and its diagonals bisect each other.
A rectangle is a parallelogram with four congruent, 90° angles and congruent diagonals.
A rhombus is a parallelogram with four congruent sides and perpendicular diagonals that bisect their angles.
A square has all of the properties of a rhombus and a rectangle: four congruent sides, four congruent, 90° angles, two pairs of parallel sides, consecutive angles that are supplementary, and diagonals that are congruent, perpendicular, bisect each other, and bisect their angles.
A trapezoid is a quadrilateral with at least one pair of parallel sides.
An isosceles trapezoid is a trapezoid with a pair of congruent sides, angles, and diagonals.

The formula for the area of any parallelogram is A = bh, where A is the area, b is the base of the parallelogram, and h is the height of the parallelogram. If the base of a parallelogram is 10 centimeters and the height is 40 centimeters, then the area is (10)(40) = 400 centimeters^{2}.
This formula can also be used to find the area of a rectangle, since a rectangle is a type of parallelogram, but the area of a rectangle is usually described as A = lw, where A is the area, l is the length of the rectangle, and w is the width of the rectangle. If the length of a rectangle is 1.2 millimeters and the width is 2.5 millimeters, then the area is (1.2)(2.5) = 3 millimeters^{2}.
The area of a square could be found with either formula, since a square is both a parallelogram and a rectangle, but it is easier to describe the area of a square as A = s^{2}, where s is the length of one side of the square, since all four sides of a square are congruent. If one side of a square is 0.17 meters, then the area of the square is (0.17)^{2} = 0.0289 meters^{2}.
The area of a trapezoid is A = (b_{1}+ b_{2})h, where A is the area, b_{1} and b_{2} are the bases of the trapezoid, and h is the height of the trapezoid. If the lengths of the bases of a trapezoid are 3 feet and 9 feet, respectively, and the height is 10 feet, then the area of the trapezoid is (3 + 9)10 = (12)(10) = (6)(10) = 60 feet^{2}.
The perimeter of any quadrilateral is the sum of the lengths of each of its four sides, but we also have separate formulas for rectangles and squares. The perimeter of a rectangle can be described as P = 2l + 2w, since the two lengths of the rectangle are the same, as are the two widths. Since all four sides of a square measure the same, the perimeter of a square can be described as P = 4s. Just multiply the length of one side by 4. If one side of a square is 5.4 centimeters, then the perimeter of the square is (5.4)(4) = 21.6 centimeters.
Just as with triangles, quadrilateral word problems about area or perimeter usually contain the word area or the word perimeter. Phrases such as distance around can describe perimeter, and phrases such as surface covered can indicate area.
Example
What is the distance around a rectangle that has a length of 2 yards and a width of 8.9 yards?
Read the entire word problem.
We are given the length and width of a rectangle.
Identify the question being asked.
We are looking for the distance around the rectangle.
Underline the keywords and words that indicate formulas.
The words distance around indicate that we are looking for the perimeter of the rectangle, which is equal to 2l + 2w.
Cross out extra information and translate words into numbers.
There is no extra information in this problem.
List the possible operations.
We will need to multiply both the length and the width by 2 and then add those products.
Write number sentences for each operation.
2(2) + 2(8.9)
Solve the number sentences and decide which answer is reasonable.
2(2) + 2(8.9) = 4 + 17.8 = 21.8 yards
Check your work.
Subtract the length of each side from the perimeter. Since the perimeter is the total distance around the rectangle, we should be left with zero after subtracting every length: 21.8 – 2 – 2 – 8.9 – 8.9 = 0
Word problems can combine more than one formula. For instance, you may use the perimeter of a square to find the length of one side of that square, and then use the length of the side to find the area of the square.
Example
If the perimeter of a square is 44 inches, what is the area of the square?
Read the entire word problem.
We are given the perimeter of a square.
Identify the question being asked.
We are looking for the area of the square.
Underline the keywords and words that indicate formulas.
The words perimeter 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 perimeter of a square is equal to 4s, so we will need to divide 44 by 4 to find the length of one side of the square. The area of a square is equal to s^{2}, so we will need to square our quotient.
Write number sentences for each operation.
First, find the length of one side of the square:
Solve the number sentences and decide which answer is reasonable.
= 11 inches
Write number sentences for each operation.
One side of the square is 11 inches, which means that the area of the square is equal to 11^{2}.
Solve the number sentences and decide which answer is reasonable.
11^{2} = 121 square inches
Check your work.
Take the square root of the area to find the length of one side of the square. Then, multiply that by 4 to find the perimeter of the square, which should equal 44 inches: √121 = 11, (4)(11) = 44 inches.
Caution
When working with a word problem that involves both area and perimeter, be sure watch your units carefully. Perimeter is found in linear units, and area is found in square units.

Pace Yourself
Using a ruler, a pencil, and paper, draw a rectangle. What is the area and perimeter of your rectangle? Now, divide your rectangle into two triangles. What is the area and the perimeter of each triangle?

Circles: Subject Review
The formula for area of a circle is A = πr^{2}, where A is the area of the circle and r is the radius of the circle. If a circle has a radius of 10 feet, then its area is π(10)^{2} = 100π feet^{2}.
Fuel for Thought
A circle is a set of connected points that are all the same distance from a single point, which is the center of the circle. A circle is a closed figure, but not a polygon.
A radius is the distance from the center of a circle to a point on the circle.
A diameter is the distance from one side of a circle to the other through the center of the circle.
The circumference of a circle is the distance around the outside of a circle.

The diameter of a circle is twice the radius, which means that d = 2r and r = d, where d is the diameter and r is the radius. If the diameter of a circle is 27 feet, then the radius of the circle is (2) = 13.5 feet. If the radius of another circle is 2.8 centimeters, then the diameter of that circle is 2(2.8) = 5.6 centimeters.
The circumference of a circle is equal to 2πr, or πd. If the diameter of a circle is 27 feet, then the circumference of that circle is 27π feet. If the circumference of a circle is 10π inches, then the diameter of the circle is 10 inches and the radius is 5 inches.
Circumference sounds a lot like perimeter, and it may appear in word problems the same way. If a word problem asks you for the distance around a circle, it's asking you for the circumference of the circle. The keywords radius and diameter will also indicate that you are working with a circle, although the word circle almost always appears in any word problem that involves a circle.
Inside Track
Since we have two formulas for the circumference of a circle, one that uses the radius and one that uses the diameter, choose which formula to use based on the information given to you in a problem. If you are given the radius, use C = 2πr, and if you are given the diameter, use C= πd. You may save yourself the step of finding the radius from the diameter, or vice versa.

Example
What is the radius of a circle whose circumference is 24π units?
Read the entire word problem.
We are given a circumference.
Identify the question being asked.
We are looking for a radius.
Underline the keywords and words that indicate formulas.
The words circumference 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 circumference is C = 2πr, so we can find the radius by dividing the circumference by 2π.
Write number sentences for each operation.
Solve the number sentences and decide which answer is reasonable.
= 12 units
Check your work.
Since C = 2πr, multiply the radius by 2π to check that it is equal to the circumference, 24π units: 2π(12) = 24π units.
Volume: Subject Review
The formula of the volume of a rectangular prism is V = lwh, where V is the volume of the prism, l is the length, w is the width, and h is the height. If a rectangular prism has a length of 6 inches, a width of 10 inches, and a height of 20 inches, then it has a volume of (6)(10)(20) = 1,200 inches^{3}.
Fuel for Thought
Volume is the amount of space taken up by a threedimensional object. Volume is always found in cubic units.

The formula of the volume of a cube, which is a type of rectangular prism, is V = e^{3}, where V is the volume of the prism and e is the length of one edge of the cube. A cube with an edge of 5 feet has a volume of (5)^{3} = 125 feet^{3}.
The formula to find the volume of a pyramid is V = bh, where V is the volume of the pyramid, b is the area of the base, and h is the height. If the pyramid has a base of 3 square feet and a height of 12 feet, then it has a volume of (3)(12) = 12 feet^{3}.
The formula to find the volume of a cylinder is V = πr^{2}h, where V is the volume of the cylinder, r is the radius, and h is the height. If a cylinder has a radius of 7 centimeters and a height of 15 centimeters, then it has a volume of π(7)^{2}(15) = 735 centimeters^{3}.
The formula for the volume of a cone is V = πr^{2}h, where V is the volume of the cone, r is the radius, and h is the height. If a cone has a radius of 15 m and a height of 30 meters, then it has a volume of π(15)^{2}(30) = 2,250 meters^{3}.
The formula for the volume of a sphere is V = πr^{3}, where V is the volume of the sphere and r is the radius. If a sphere has a radius of 3 inches, then it has a volume of π(3)^{3} = 36π inches^{3}.
No tricks to a volume word problem—volume word problems almost always use the word volume and give the measurements of the type of solid whose volume you must find. Since volume is the amount of space an object occupies, you may be asked for the amount of space taken up instead of the volume of a solid. That phrase should always signal volume.
Example
If a cube has an edge of 5 centimeters, how much space does the cube take up?
Read the entire word problem.
We are given the edge of a cube.
Identify the question being asked.
We are looking for how much space it takes up.
Underline the keywords and words that indicate formulas.
The phrase how much space does the cube take up asks us to find the volume 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 volume of a cube is V = e^{3}, so we can find the volume by cubing, or multiplying by itself three times, the edge of the cube.
Write number sentences for each operation.
5^{3}
Solve the number sentences and decide which answer is reasonable.
5^{3} = 125 centimeters^{3}
Check your work.
Since V = e^{3}, we can find the length of the edge by taking the cube root of the volume. The cube root of 125 should equal 5 centimeters: 3√125 = 5 centimeters.
Caution
Volume word problems may be easy to identify, but be sure to select the right volume formula. The volume formulas for pyramids, cylinders, cones, and spheres are very similar, but are, in fact, a little different from one another. Determine which solid you are working with, and then carefully apply the right formula.

Pace Yourself
Find a cube, a rectangular prism, and a cylinder in your home. A box of tissues may be a cube, a shoebox may be a rectangular prism, and a roll of paper towels may be a cylinder. Measure each with a ruler and find the volume of each. Which object has the largest volume?

Surface Area: Subject Review
We can also find the surface area of solid figures.
Fuel for Thought
Surface area is the sum of the areas of each face, or side, of a solid figure.

The surface area of a rectangular prism is SA = 2(lw + wh + lh), where SA is the surface area, l is the length, w is the width, and h is the height. If the length of a prism is 1.5 meters, the width is 2.8 meters, and the height is 0.5 meters, then the surface area is equal to 2[(1.5)(2.8) + (2.8)(0.5) + (1.5)(0.5)] = 2(4.2 + 1.4 + 0.75) = 2(6.35) = 12.7 meters^{2}.
The surface area of a cube is SA = 6s^{2}, where SA is the surface area and s is the length of one side of the cube. If one side of the cube measures 14 centimeters, then the surface area of the cube is 6(14)^{2} = 6(196) = 1,176 centimeters^{2}.
The surface area of a cylinder is SA = 2πr^{2} + 2πrh, where SA is the surface area, r is the radius, and h is the height. A cylinder with a radius of 8 millimeters and a height of 4 millimeters has a surface area of 2π(8)^{2} + 2π(8)(4) = 2π(64) + 2π(32) = 128π + 64π = 192π millimeters^{2}.
The surface area of a sphere is SA = 4πr^{2}, where SA is the surface area and r is the radius. A sphere with a radius of 2 feet has a surface area of 4π(2)^{2} = 4π(4) = 16π feet^{2}.
A word problem may describe the surface area as the area of every face of a solid or the total area of a solid. We will plug the given values into one of the four surface area formulas to find the surface area of the given solid. We may be given the area of one or more sides of the solid, or the volume of the solid. We can use those values to find the dimensions of the solid, and then find the surface area of the solid.
Example
If the area of one side of a cube is 80 centimeters^{2}, what is the total area of the cube?
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 total area of the cube.
Underline the keywords and words that indicate formulas.
The phrase total area means that we are looking for the surface area of the cube.
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 of one side of a cube, and a cube has six sides.
The surface area of the cube is equal to six times the area of one side.
Write number sentences for each operation.
(6)(80)
Solve the number sentences and decide which answer is reasonable.
(6)(80) = 480 centimeters^{2}
Check your work.
Divide the surface area by 6, and we should get the given area of one side of the cube, 80 centimeters^{2}: = 80 centimeters^{2}.
Inside Track
The formulas for volume and surface area of some solids can be complex. Remember that the exponent of the radius is always two in surface area formulas and three in volume formulas.

Summary
Most geometry problems give you a diagram, but now you're prepared to handle a geometry problem made up only of words. And at the same time, we've reviewed many important geometry rules and formulas.
Find practice problems and solutions for these concepts at Geometry Word Problems Practice Problems.