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# Volume

## Number of unit cubes to fill prisms, spheres, pyramids, cylinders, cones, and composite solids.

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Volume of Solids

The composite solid below is made of a cube and a square pyramid. The length of each edge of the cube is 12 feet and the overall height of the solid is 22 feet. What is the volume of the solid? Why might you want to know the volume of the solid?

### Volume of a Solid

The volume of a solid is the number of unit cubes it takes to fill up the solid.

A prism is a solid with two congruent polygon bases that are parallel and connected by rectangles. Prisms are named by their base shape.

To find the volume of a prism, find the area of its base and multiply by its height.

Prism:V=ABaseh\begin{align*}Prism: V=A_{Base} \cdot h\end{align*}

A cylinder is like a prism with a circular base.

To find the volume of a cylinder, find the area of its circular base and multiply by its height.

Cylinder:V=πr2h\begin{align*}Cylinder: V=\pi r^2 h \end{align*}

A cone also has a circular base, but its lateral surface meets at a point called the vertex.

To find the volume of a cone, find the volume of the cylinder with the same base and divide by three.

Cone:V=πr2h3\begin{align*}Cone: V=\frac{\pi r^2 h}{3}\end{align*}

A pyramid is similar to a cone, except it has a base that is a polygon instead of a circle. Like prisms, pyramids are named by their base shape.

To find the volume of a pyramid, find the volume of the prism with the same base and divide by three.

Pyramid:V=ABaseh3\begin{align*}Pyramid: V= \frac{A_{Base} \cdot h}{3}\end{align*}

A sphere is the set of all points in space equidistant from a center point. The distance from the center point to the sphere is called the radius.

The volume of a sphere relies on its radius.

Sphere:V=43πr3\begin{align*}Sphere: V= \frac{4}{3} \pi r^3\end{align*}

A composite solid is a solid made up of common geometric solids.

The volume of a composite solid is the sum of the volumes of the individual solids that make up the composite.

Let's look at some problems where we find the volume.

1. Find the volume of the rectangular prism below.

To find the volume of the prism, you need to find the area of the base and multiply by the height. Note that for a rectangular prism, any face can be the “base”, not just the face that appears to be on the bottom.

Area of BaseHeightVolume=44=16 in2=5 in=80 in3\begin{align*}Area \ of \ Base &= 4 \cdot 4 = 16 \ in^2 \\ Height &= 5 \ in \\ Volume &=80 \ in^3\end{align*}

2. Find the volume of the cone below.

To find the volume of the cone, you need to find the area of the circular base, multiply by the height, and divide by three.

Area of CircleVolume=72π=49π cm2=49π123=196π cm3\begin{align*}Area \ of \ Circle &= 7^2 \pi = 49 \pi \ cm^2 \\ Volume &= \frac{49 \pi \cdot12}{3} = 196 \pi \ cm^3\end{align*}

3. Find the volume of a sphere with radius 4 cm.

The formula for the volume of a sphere is V=43πr3\begin{align*}V= \frac{4}{3} \pi r^3\end{align*}.

Volume=43π(4)3=256π3cm3\begin{align*}Volume= \frac{4}{3} \pi (4)^3 = \frac{256 \pi}{3} cm^3\end{align*}

### Examples

#### Example 1

Earlier, you were asked what is the volume of the solid, and why might you want to know the volume of the solid.

The composite solid below is made of a cube and a square pyramid. The length of each edge of the cube is 12 feet and the overall height of the solid is 22 feet.

To find the volume of the solid, find the sum of the volumes of the prism (the cube) and the pyramid. Note that since the overall height is 22 feet and the height of the cube is 12 feet, the height of the pyramid must be 10 feet.

Volume of Prism(Cube)Volume of PyramidTotal Volume=ABaseh=(1212)12=1728 ft3=ABaseh3=(1212)103=480 ft3=1728+480=2208 ft3\begin{align*}Volume \ of \ Prism(Cube) &=A_{Base} \cdot h=(12 \cdot 12) \cdot 12=1728 \ ft^3 \\ Volume \ of \ Pyramid &=\frac{A_{Base} \cdot h}{3}=\frac{(12 \cdot 12) \cdot 10}{3}=480 \ ft^3 \\ Total \ Volume &=1728+480=2208 \ ft^3\end{align*}

The volume helps you to know how much the solid will hold. One cubic foot holds about 7.48 gallons of liquid. This solid would hold 16,515.84 gallons of liquid!

#### Example 2

The area of the base of the pyramid below is 100 cm2\begin{align*}100 \ cm^2\end{align*}. The height is 5 cm. What is the volume of the pyramid?

V=100 cm25 cm=500 cm3\begin{align*}V=100 \ cm^2 \cdot 5 \ cm=500 \ cm^3\end{align*}

#### Example 3

The volume of a sphere is 500π3 in3\begin{align*}\frac{500 \pi}{3} \ in^3\end{align*}. What is the radius of the sphere?

43πr34r3r3r=500π3=500=125=5 in\begin{align*}\frac{4}{3}\pi r^3 &=\frac{500 \pi}{3} \\ 4r^3 &=500 \\ r^3 &=125 \\ r &=5 \ in\end{align*}

#### Example 4

The volume of a square pyramid is 64 in3\begin{align*}64 \ in^3\end{align*}. The height of the pyramid is three times the length of a side of the base. What is the height of the pyramid?

64=ABaseh3=s2h3=s2(3s)3=s3\begin{align*}64=\frac{A_{Base} \cdot h}{3}=\frac{s^2h}{3}=\frac{s^2(3s)}{3}=s^3\end{align*}. Therefore, s=4 in\begin{align*}s=4 \ in\end{align*} and h=3(4)=12 in\begin{align*}h=3(4)=12 \ in\end{align*}.

### Review

Find the volume of each solid or composite solid.

1.

2.

3.

4.

5. The base is an equilateral triangle.

6.

7. Explain why the formula for the volume of a prism involves the area of the base.

8. How is a cylinder related to a prism?

9. How is a pyramid related to a cone?

10. How is a sphere related to a circle?

11. If one cubic centimeter will hold 1 milliliter of water, approximately how many liters of water will the solid in #1 hold? (One liter is 1000 milliliters).

12. If one cubic centimeter will hold 1 milliliter of water, approximately how many liters of water will the solid in #3 hold? (One liter is 1000 milliliters).

13. If 231 cubic inches will hold one gallon of water, approximately how many gallons of water will the solid in #5 hold?

14. The volume of a cone is 125π in3\begin{align*}125 \pi \ in^3\end{align*}. The height is three times the length of the radius. What is the height of the cone?

15. The volume of a pentagonal prism is 360 in3\begin{align*}360 \ in^3\end{align*}. The height of the prism is 3 in. What is the area of the pentagon base?

To see the Review answers, open this PDF file and look for section 1.10.

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### Vocabulary Language: English

Composite

A number that has more than two factors.

Cone

A cone is a solid three-dimensional figure with a circular base and one vertex.

Cylinder

A cylinder is a solid figure with two parallel congruent circular bases.

Prism

A prism is a three-dimensional object with two congruent parallel bases that are polygons.

Pyramid

A pyramid is a three-dimensional object with a base that is a polygon and triangular faces that meet at one vertex.

Sphere

A sphere is a round, three-dimensional solid. All points on the surface of a sphere are equidistant from the center of the sphere.

Volume

Volume is the amount of space inside the bounds of a three-dimensional object.