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Solids

Calculating Volumes of Solids

Concept

Volume

The volume of a solid is the measure of the amount of space inside the solid. It is the three-dimensional equivalent of the area. The volume is measured using cubic units, such as cubic meters, The applet below illustrates the volume of some solids as a blue region by pressing the corresponding buttons.
An applet that illustrates the volume of a cube, a cone, and a sphere by fillining them.
Different solids may have different formulas to calculate their volumes.
Figure Name Formula
A cube with its sides labeled
Cube
A cone with radius r and height h
Cone
A cylinder with radius r and height h
Cylinder
A shpere with radius r
Sphere

Rule

Cavalieri's Principle

Two solids with the same height and the same cross-sectional area at every altitude have the same volume. This means that, as long as their heights are equal, skewed versions of the same solid have the same volume.
If and are the volumes of the above solids, then they are equal.

Proof

Informal Justification

This principle will be proven by using a set of identical coins. Consider a stack in which each of these coins is placed directly on top of each other. Consider also another stack where the coins lie on top of each other, but are not aligned.

stack

The first stack can be considered as a right cylinder. Similarly, the second stack can be considered as an oblique cylinder, which is a skewed version of the first cylinder. Because the coins are identical, the cross-sectional areas of the cylinders at the same altitude are the same.

stack

Since the coins are identical, they have the same volume. Furthermore, since the height is the same for both stacks, they both have the same number of coins. Therefore, both stacks — cylinders — have the same volume. This reasoning is strongly based on the assumption that the face of the coins have the same area.

Rule

Volume Formulas

The volume of a solid can be calculated with a formula that depends on the solid's shape.

Rule

Prism

The volume of a prism is given by multiplying the area of its base, with its height,

Rule

Pyramid

The volume of a pyramid is a third of the volume of a prism with the same base. Thus, the formula for the pyramid's volume is:

Rule

Cylinder

A cylinder has a base of a circle. Therefore, the cylinder's volume is calculated by multiplying the area of the circle with the height of the cylinder.

Proof

Volume of a Cylinder

Proof

Volume of a Cylinder

The formula for the volume of a cylinder can be proven by placing a rectangular prism next to it.

Cuboid and Cylinder.svg

Suppose the prism and the cylinder have the same base area and height.

Cuboid and Cylinder2.svg

Thus, Cavalieri's principle states that two solids with the same base area and height have the same volume. Since the volume of a prism is given by the volume of a cylinder can be calculated with the same formula. The base of a cylinder has the shape of a circle. Its area can be expressed as Therefore, the formula for the volume of a cylinder is

It applies to all cylinders because there is always a prism with the same base area and height.

Rule

Cone

The volume of a cone is a third of the volume of a cylinder with the same base. Since the cone's base is in the shape of a circle, the formula for the volume of the cone is:

Proof

Volume of a Cone

Proof

Volume of a Cone

The formula for the volume of a cone can be derived using a pyramid.

Pyramid cone2.svg

Assume, the pyramid and cone have the same height, and that the pyramid has a square base, whose side length is twice the radius of the cone's base.

Pyramid cone.svg

Since the cone's diameter is equal to the side length of the square, the cone will fit inside the pyramid.

Pyramid cone3.svg

The base of the cone is a circle that fits exactly in the base of the pyramid. By studying the figure from below, the ratio between the bases can be determined.

Pyramid cone4.svg
The area of the square is calculated by taking the square of the side length. Further, the area of the circle is given by the square of the radius times The ratio between the areas can now be calculated by dividing the area of the circle with the area of the square.
The ratio between the areas is This will be the case for all cross sections of the pyramid and cone. Therefore, Cavalieri's principle implies that the volume of the pyramid is equal to the volume of the cone scaled by the factor The formula for the volume of a pyramid is: As mentioned previously, the base area for the pyramid is expressed as By multiplying the volume of the pyramid with the scale factor, the formula for the volume of the cone can be found.
The result is the formula for the volume of a cone.

Rule

Sphere

A sphere is a perfectly round solid composed of many circles. The formula for the volume of the sphere is given by: where is the radius of the largest circle in the sphere.

Concept

Composite Solid

A solid that is made up of more than one solid is called a composite solid. The individual solids can be combined either by adding or subtracting them from one another. For instance, a hemisphere — half a sphere — can be combined with a cone to make something that resembles a snow cone, or it could be used to hollow out a cylinder.

Concept Composite Solid.svg
The volume of a composite solid is either the sum or difference between the volumes of the individual solids, whichever is applicable.
fullscreen
Exercise

Find the volume of ice cream in the wafer cup.

IceCreamSolid.svg

Assume that the wafer is in the shape of a cone and that the ice cream above the wafer is a hemisphere.

Show Solution
Solution

The ice cream can be interpreted as a composite solid, a combination of a cone and a hemisphere.

IceCreamSolid2.svg
Thus, the sum of the cone's volume and the hemisphere's volume will be the volume of the ice cream. Let's start with calculating the volume of the cone.

Theory

Volume of the Cone

The volume of a cone can be calculated with the formula where is the area of the base and is the height. The base of a cone is in the shape of a circle, Its area can be calculated using The radius of the base is the same as for the hemisphere, since it fits exactly with the cone.
IceCreamSolid3.svg
The height of the cone is inch and the radius is inch. By substituting the known values, the volume of the cone can be calculated.
Evaluate right-hand side
The volume of the cone is .

Theory

Volume of the Hemisphere

A hemisphere is half of a sphere. Therefore, to calculate the volume we can use the formula for the volume of a sphere and multiply it by The formula requires the radius of the hemisphere, which is inch.
IceCreamSolid4.svg
Now, the volume of the hemisphere can be calculated.
Evaluate right-hand side
Thus, the volume of the hemisphere is

Theory

Volume of the Composite Solid

The volume of the composite solid can be found by adding the volume of the cone, and the volume of the hemisphere,
Evaluate right-hand side
Therefore, the volume of ice cream in the wafer is
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