11.12 Spheres

The volume and surface area of a sphere are given by the formulas in terms of the radius r which are V=\dfrac{4}{3}\pi r^3 and SA=4\pi r^2, respectively.

Using these formulas, we can find the surface area and volume using the radius.

In addition to this, knowing the surface area of a sphere can allow us to find its radius, which can then be used to find the volume. In the same way, we can find the surface area of a sphere if we are given its volume.

To find the surface areas and volumes of various parts of the sphere, we can think of them as fractions of the sphere.

Since a hemisphere is equal to half a sphere, it's volume will be equal to half the volume of a sphere with the same radius. This tells us that the volume of a hemisphere is given by the formula:\text{Volume of a hemisphere}=\dfrac{2}{3}\pi r^3

Similarly, the curved surface of the hemisphere will have an area equal to half the surface area of a sphere. Since the circular base of the hemisphere has an area of \pi r^2, the total surface area of a hemisphere is given by the formula:\text{Surface area of a hemisphere}=3\pi r^2

We can use the same types of calculations to find the volume and surface area of other fractions of the sphere.

Examples

Example 1

Find the surface area of the sphere shown. Round your answer to two decimal places.

Worked Solution

Create a strategy

Use the surface area formula SA=4\pi r^2.

Apply the idea

We are given r=9.

\displaystyle SA	\displaystyle =	\displaystyle 4\pi r^2	Write the formula
	\displaystyle =	\displaystyle 4\times\pi \times 9^2	Substitute r
	\displaystyle \approx	\displaystyle 1017.88 \text{ cm}^2	Evaluate and round

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Example 2

Find the volume of the sphere shown. Round your answer to two decimal places.

Worked Solution

Create a strategy

Use the volume of a sphere formula: V=\dfrac{4}{3}\pi r^3.

Apply the idea

\displaystyle V	\displaystyle =	\displaystyle \dfrac{4}{3}\pi r^3	Write the formula
	\displaystyle =	\displaystyle \dfrac{4}{3}\times\pi \times \left(4\right)^3	Substitute r
	\displaystyle \approx	\displaystyle 268.08 \text{ cm}^3	Evaluate and round

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Example 3

A sphere has a radius r\text{ cm} long and a volume of \dfrac{512\pi}{3}\text{ cm}^3.

Find the radius of the sphere. Round your answer to two decimal places.

Worked Solution

Create a strategy

Use the volume formula V=\dfrac{4}{3}\pi r^3 to find the radius.

Apply the idea

\displaystyle V	\displaystyle =	\displaystyle \dfrac{4}{3}\pi r^3	Write the formula
\displaystyle \dfrac{512\pi}{3}	\displaystyle =	\displaystyle \dfrac{4}{3}\pi r^3	Substitute the volume
\displaystyle 512\pi	\displaystyle =	\displaystyle 4\pi r^3	Multiply both sides by 3
\displaystyle \dfrac{512\pi}{4\pi}	\displaystyle =	\displaystyle r^3	Divide both sides by 4\pi
\displaystyle r^3	\displaystyle =	\displaystyle 128	Simplify
\displaystyle r	\displaystyle =	\displaystyle \sqrt[3]{128}	Cube root both sides
	\displaystyle \approx	\displaystyle 5.04\text{ cm}	Evaluate and round

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