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Geometry Word Problems Study Guide (page 4)

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Updated on Aug 24, 2011

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 inches3.

Fuel for Thought

Volume is the amount of space taken up by a three-dimensional 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 = e3, 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 feet3.

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 feet3.

The formula to find the volume of a cylinder is V = πr2h, 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 centimeters3.

The formula for the volume of a cone is V = πr2h, 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 meters3.

The formula for the volume of a sphere is V = πr3, 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 inches3.

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 = e3, 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.

53

Solve the number sentences and decide which answer is reasonable.

53 = 125 centimeters3

Check your work.

Since V = e3, 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 meters2.

The surface area of a cube is SA = 6s2, 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 centimeters2.

The surface area of a cylinder is SA = 2πr2 + 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 millimeters2.

The surface area of a sphere is SA = 4πr2, 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 feet2.

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 centimeters2, 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 centimeters2

Check your work.

Divide the surface area by 6, and we should get the given area of one side of the cube, 80 centimeters2: = 80 centimeters2.

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.

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