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Stage 5.1-3

11.12 Spheres

Lesson

Introduction

A sphere with centre and radius drawn on it.

A sphere is a perfectly round object with one curved face

Surface area and volume of a sphere

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.

A hemisphere

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.

A sphere with radius of 9 centimetres.
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^2Write the formula
\displaystyle =\displaystyle 4\times\pi \times 9^2Substitute r
\displaystyle \approx\displaystyle 1017.88 \text{ cm}^2Evaluate and round

Example 2

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

A sphere with a radius of 4 centimetres.
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^3Write the formula
\displaystyle =\displaystyle \dfrac{4}{3}\times\pi \times \left(4\right)^3Substitute r
\displaystyle \approx\displaystyle 268.08 \text{ cm}^3Evaluate and round

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

Surface area and volume of the sphere:

\displaystyle \begin{aligned} SA&=4\pi r^2\\ V&=\dfrac{4}{3}\pi r^3 \end{aligned}
\bm{r}
is the radius of the sphere

Outcomes

MA5.3-13MG

applies formulas to find the surface areas of right pyramids, right cones, spheres and related composite solids

MA5.3-14MG

applies formulas to find the volumes of right pyramids, right cones, spheres and related composite solids

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