Inroduction to Volume Word Problems
… treat Nature by the sphere, the cylinder and the cone …
—PAUL CÉZANNE (1839–1906)
This lesson will explain each of the volume formulas for common geometric solids, and how to apply these formulas to word problems with these figures.
Volume
The volume of a threedimensional figure is the amount of space in the figure.
Many volume formulas can be summarized as the area of the base multiplied by the height, or V = Bh.
Tip:
The volume of any figure is expressed in cubic units, or units^{3}.

Volume of a Rectangular Prism
A rectangular prism is a threedimensional solid whose faces are all rectangles. The formula for the volume of a rectangular solid is Volume = length × width× height, or V = l × w × h.
Example
What is the volume of a rectangular prism with a height of 10 m, a width of 14 m, and a height of 20 m?
Read and understand the question. This question is looking for the volume of a rectangular prism when the three dimensions are given.
Make a plan. Use the formula V = l × w × h, and substitute the given values. Then, evaluate the formula.
Carry out the plan. The volume is V = 10 × 14 × 20 = 2,800 m^{3}
Check your answer. To check this solution, divide the volume by two of the dimensions to check if the result is the third dimension: 2,800 ÷ 10 = 280; 280 ÷ 14 = 20, which is the remaining dimension. This answer is checking.
Volume of a Cube
A cube is a special type of rectangular prism where each face is the same shape and size. A cube has faces that are all congruent squares, so each edge is the same length. The volume of a cube can be found by using the formula for rectangular prisms, but it can also be found by using the formula V = e^{3}, where e is the measure of an edge of the cube.
Example
The measure of the edge of a cube is 7 cm. What is the volume of the cube?
Read and understand the question. This question is looking for the volume of a cube when the measure of an edge of the cube is known.
Make a plan. Use the formula V = e^{3}, and substitute the given value of e. Then, evaluate the formula.
Carry out the plan. The volume formula becomes V= (7)^{3}= 7 × 7 × 7 = 343 cm^{3}.
Check your answer. To check this solution, divide the volume by the measure of the edge of the cube two times to see if the result is also the measure of the edge:
which is the measure of the edge of the cube. This answer is checking.
Volume of a Triangular Prism
A triangular prism is a threedimensional solid with triangles as the bases and rectangles as the lateral faces. An example of a triangular prism is shown next.
The formula for the volume of a triangular prism can be found by using the formula V = Bh, where B is the area of the base and h is the height of the prism.
Example
What is the volume of a triangular prism with a base area of 12 m^{2}and a height of 4 m?
Read and understand the question. This question is asking for the volume of a triangular prism when the area of each base and the height of the prism are given.
Make a plan. Use the formula V = Bh, where B is the area of the base and h is the height of the prism. Substitute the known values and evaluate the formula.
Carry out the plan. The formula becomes V= 12 × 4 = 48. The volume is 48 m^{3}.
Check your answer. To check this problem, divide the volume of the prism by the height, and check to see if the result is the area of the base: 48 ÷ 4 = 12, which is the area of the base. This answer is checking.
Volume of a Pyramid
A pyramid is a geometric solid with a polygon as a base and triangles as each of the other faces. A square base pyramid is shown in the following figure.
The volume of a pyramid is equal to onethird of the volume of a prism with the same dimensions. Therefore, the formula for the volume of a pyramid is equal to V = Bh, where B is the area of the base and h is the height of the pyramid.
Example
What is the volume of a square base pyramid with a base area of 25 cm^{2}and a height of 15 cm?
Read and understand the question. This question is looking for the area of a square base pyramid when the area of the base and the height of the pyramid are known.
Make a plan. Use the formula V = Bh, where B is the area of the base and h is the height of the pyramid. Substitute the values and evaluate the formula.
Carry out the plan. The formula becomes
V = (25)(15)
= (375) = 125
The volume is 125 cm^{3}.
Check your answer. To check your answer, work backward. Multiply the volume by 3, and then divide by one of the dimensions to check to see if the result is the other dimension.
This result is checking.
Solids With Curved Surfaces
The next two solids have curved surfaces, and each has one or two bases that are circles.
Tip:
Remember that the area of a circle is A = πr^{2}. This formula is used to find the volume of the next two solids with bases in the shape of circles.

Volume of a Cylinder
As mentioned in the previous lesson, a cylinder is a solid with two circles as the bases. To find the volume of a cylinder, multiply the area of the base by the height of the cylinder. Because the base is a circle, the formula is V = πr^{2}h, where r is the radius of the base and h is the height of the cylinder.
Example
The radius of the base of a cylinder is 6 m and the height of the cylinder is 9 m. What is the volume of the cylinder in terms of π?
Read and understand the question. This question is looking for the volume of a cylinder when the radius of the base and the height are known.
Make a plan. Substitute into the volume formula, and then evaluate to find the volume.
Carry out the plan. The formula becomes V = π(6)^{2}(9). Evaluate the exponent to get V = 36(9)π. Multiply to simplify. The volume is 324π m^{3}.
Check your answer. To check this solution, use the strategy of working backward and divide the volume by the height times π. Then, take the positive square root to see if this result is the radius of the base: 324π ÷ 9π = 36. The positive square root of 36 is 6, which is the radius of the base. This answer is checking.
Volume of a Cone
A cone is a solid with one base that is a circle. An example of a cone is shown in the following figure.
The volume of a cone is equal to onethird of the volume of a cylinder with the same height and the same size circular base. Therefore, the formula for the volume of a cone is V = πr^{2}h.
Example
What is the volume of a cone with a base radius of 9 in. and a height of 5 in.?
Read and understand the question. This question is asking for the volume of a cone when the radius of the base and the height are given.
Make a plan. Use the formula V = πr^{2}h and substitute the given values. Evaluate the formula to find the volume.
Carry out the plan. The formula becomes V = π(9)^{2}(5). Evaluate the exponent to simplify to V = (81)(5)π. Multiply. V = (405)π. Divide 405 by 3 to simplify: V = 135π in.^{3}.
Check your answer. To check this answer, work backward by multiplying the volume by 3. Then, divide by the height times π and take the positive square root of the result to get the length of the base radius:
The positive square root of 81 is 9, so this answer is checking.
Volume of a Sphere
The volume of a sphere can be found by using the formula V = πr^{3}, where r is the radius of the sphere.
Example
What is the volume of a sphere with a radius of 6 in.?
Read and understand the question. This question is looking for the volume of a sphere when the radius is given.
Make a plan. Substitute the value of r into the formula and evaluate to find the volume.
Carry out the plan. The formula becomes V = π(6)^{3}. Evaluate the exponent to get V = π(216). Multiply to get V = 288π in.^{3}.
Check your answer. To check this result, divide the volume by π. Then, see if the result is the same as 6^{3}.
288π ÷ π = 216
6^{3}is also equal to 216, so this result is checking.
Find practice problems and solutions for these concepts at Volume Word Problems Practice Questions.
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From Math Word Problems in 15 Minutes A Day. Copyright © 2009 by LearningExpress, LLC. All Rights Reserved.