 AustraliaNSW
Stage 5.1-3

# 7.08 Spheres

Lesson

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 $r$r:

Surface area of a sphere

$SA=4\pi r^2$SA=4πr2

Volume of a sphere

$V=\frac{4}{3}\pi r^3$V=43πr3

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.

#### Worked example

A sphere has a surface area of $20$20 cm2. What is the volume of the sphere?

Think: We can substitute the surface area into the equation $SA=4\pi r^2$SA=4πr2 and solve to find the radius $r$r. We can then substitute that radius into the equation $V=\frac{4}{3}\pi r^3$V=43πr3 to find the volume.

Do: If we substitute our given surface area into the surface area formula, we get:

$20=4\pi r^2$20=4πr2

Dividing both sides by $4\pi$4π and then taking the square root of both sides isolates $r$r, giving us:

$r=\sqrt{\frac{20}{4\pi}}$r=204π

Substituting this value into the volume formula gives us:

$V=\frac{4}{3}\pi\left(\sqrt{\frac{20}{4\pi}}\right)^3$V=43π(204π)3

Evaluating this and rounding to two decimal places tells us that the volume of the sphere is equal to $8.41$8.41 cm3.

Reflect: We rearranged the surface area equation to find the radius, which we then substituted into the volume equation. We kept the radius as an exact value until the final step to avoid any rounding errors.

### Parts of spheres

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

#### Exploration

Consider a hemisphere with a radius $r$r. 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:

$\text{Volume of a hemisphere}=\frac{2}{3}\pi r^3$Volume of a hemisphere=23πr3

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 $\pi r^2$πr2, the total surface area of a hemisphere is given by the formula:

$\text{Surface area of a hemisphere}=3\pi r^2$Surface area of a hemisphere=3πr2

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

#### Practice questions

##### Question 1

Find the surface area of the sphere shown. ##### Question 2

Find the volume of the sphere shown. ##### Question 3

Fill in the blank to complete the statement below:

1. If the radius of a sphere doubles, then its volume increases by a factor of $\editable{}$.

### 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