Volume of three dimensional objects

Volume of three dimensional objects

The formulae for calculating volume of prisms, cylinders and pyramids

We have already seen formulas for calculating the surface areas of some three-dimensional figures. Now, let us also see the formulas for calculating the volumes of such figures.

The amount of space that is enclosed by a space figure is called the volume. Volume is measured in cubic units, such as cubic meters (m³), cubic centimeters (cm³), etc. When we calculate the volume of a space figure or solid, we are finding the number of cubic units enclosed by the given space figure.

Volume of a right prism

The figure shows a right rectangular prism. Let $h$ be the height, $w$ the width, and $\ell$ the length of the prism. Then, the volume of the prism is given by:

$$V = \text{Base area} \times \text{height}$$

or

$$V = \ell \times w \times h$$

Generally, the volume of any right prism is equal to the product of the area of the base and the height:

$$V = \text{Base area} \times \text{height}$$

Volume of a right cylinder

Consider a right circular cylinder with radius $r$ and height $h$. The volume of a right circular cylinder is equal to the product of the area of the base and the height. If $V$ is the volume, $A$ is the area of the base, and $h$ is the height, then:

$$V = A \times h$$

Since the base is a circle, $A = \pi r^2$, so:

$$V = \pi r^2 h$$

Volume of a pyramid

Generally, the volume of a pyramid is one-third the product of its altitude (height) and its base area. If $h$ is the perpendicular distance from the vertex of the pyramid to its base, then:

$$V = \frac{1}{3} \times \text{Base area} \times h$$

Volume of a cone

Consider a cone of radius $r$ and altitude $h$. The volume of a cone is given by:

$$V = \frac{1}{3} \pi r^2 h$$

Volume of a sphere

The figure shows a sphere of radius $r$. If the sphere is placed inside a cylinder of the same radius $r$, then the height $h = 2r$. The volume of the sphere is given by:

$$V = \frac{4}{3} \pi r^3$$

Summary

• Volume of a prism: $V = \text{Base area} \times \text{height}$
• Volume of a cylinder: $V = \pi r^2 h$
• Volume of a pyramid: $V = \frac{1}{3} \times \text{Base area} \times h$
• Volume of a cone: $V = \frac{1}{3} \pi r^2 h$
• Volume of a sphere: $V = \frac{4}{3} \pi r^3$