Lesson 10-5: Volume of Composite Solids

Property

Cube whose edge is of length $L$:

Volume $= L^3$

Property

A cylinder's volume is found using $V = Bh$. Since the base 'B' is a circle with area $B = \pi r^2$, the specific formula for a cylinder becomes $V = \pi r^2 h$.

Examples

A can with radius 3 cm and height 10 cm has a volume of $V = \pi (3 \text{ cm})^2(10 \text{ cm}) = 90\pi \text{ cm}^3$.
A battery with a diameter of 14 mm (radius 7 mm) and height 50 mm has a volume of $V \approx \frac{22}{7} (7 \text{ mm})^2(50 \text{ mm}) = 7700 \text{ mm}^3$.

Explanation

Think of a can of Pringles. First, find the area of a single chip (the base 'B') using $\pi r^2$. Then, just multiply that base area by the height of the can 'h' to find its total volume. Remember to use the radius, which is half the diameter, to get the right answer!

Property

The volume $V$ of a cone with radius $r$ and height $h$ is given by the formula:

Examples

• A cone with a radius of $3$ cm and a height of $5$ cm has a volume of $V = \frac{1}{3}\pi (3^2)(5) = 15\pi$ cm$^3$.
• A cone with a radius of $4$ inches and a height of $9$ inches has a volume of $V = \frac{1}{3}\pi (4^2)(9) = 48\pi$ inches$^3$.

Explanation

The volume of a cone measures the amount of space it occupies. This formula shows that the volume depends on the radius of its circular base ($r$) and its perpendicular height ($h$). An important relationship to note is that a cone''s volume is exactly one-third the volume of a cylinder with the same radius and height. To calculate the volume, substitute the known values for the radius and height into the formula.

Property

The volume of a sphere is given by

where $r$ is the radius of the sphere. Recall that $r^3$, which we read as '$r$ cubed,' means $r \times r \times r$.

Examples

• A gumball has a radius of 1 centimeter. Its volume is $V = \frac{4}{3} \pi (1)^3 = \frac{4}{3}\pi \approx 4.19$ cubic centimeters.
• A soccer ball has a diameter of 22 cm, so its radius is 11 cm. Its volume is $V = \frac{4}{3} \pi (11)^3 = \frac{4}{3} \pi (1331) \approx 5575.28$ cubic centimeters.
• A spherical ornament has a volume of $36\pi$ cubic inches. To find its radius, solve $36\pi = \frac{4}{3}\pi r^3$, which simplifies to $27 = r^3$, so the radius is $r = \sqrt[3]{27} = 3$ inches.

Explanation

Volume measures the space inside a 3D shape, like a ball or a planet. For a sphere, you cube the radius (multiply it by itself three times), then multiply by pi ($\pi$), and finally multiply by the fraction $\frac{4}{3}$.