Volume of Cylinders and Spheres Revision Notes for HSC SSCE Mathematics Standard

Volume of Cylinders and Spheres

Understanding cylinders

A cylinder is a three-dimensional shape that has two circular ends (called bases) connected by straight parallel sides. You can think of it as a can or a tube. The cylinder has a circular cross-section, which means if you slice through it horizontally, you always see a circle.

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The key measurements for a cylinder are:

Radius $(r)$ : the distance from the centre to the edge of the circular base
Height $(h)$ : the vertical distance between the two circular ends

The volume formula for cylinders

To find the volume of a cylinder, we need to calculate how much space it occupies. This is done by multiplying the area of the circular base by the height of the cylinder.

Since the base is a circle with area $A = \pi r^2$ , we can write:

$V = A \times h$

Substituting the circle area formula:

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

This simplifies to:

$V = \pi r^2 h$

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This formula tells us that the volume of a cylinder is the product of $\pi$ , the square of the radius, and the height. It combines the familiar circle area formula $(\pi r^2)$ with the height to give us a three-dimensional measurement.

Worked example: Finding the volume of a cylinder

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Worked Example: Basic Cylinder Volume

Question: A cylinder has a radius of $8$ mm and a height of $12$ mm. Find the volume of the cylinder, correct to two decimal places.

Solution:

Begin with the volume formula for a cylinder:

$V = \pi r^2 h$

Replace $r = 8$ and $h = 12$ into the formula:

$V = \pi \times 8^2 \times 12$

Calculate the value:

$V = \pi \times 64 \times 12$

$V = 768\pi$

$V = 2412.743158...$

Round the answer to two decimal places and include the correct units:

$V \approx 2412.74 \text{ mm}^3$

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Exam tip: Always remember to cube the units when finding volume. If the measurements are in mm, the volume will be in mm³.

Worked example: Finding the volume of an annulus prism

An annulus is a ring shape - a circle with a circular hole in the middle. An annulus prism is this ring shape extended to create a three-dimensional object.

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Worked Example: DVD Annulus Prism

Question: The diameter of a DVD is $12$ cm, and the diameter of the hole in its centre is $1.5$ cm. Find its volume in cubic centimetres, correct to two decimal places, given that the DVD is $0.12$ cm thick.

Solution:

First, find the area of the annulus (the ring shape) using the formula $A = \pi(R^2 - r^2)$ where $R$ is the outer radius and $r$ is the inner radius.
The outer radius is $R = 12 \div 2 = 6$ cm
The inner radius is $r = 1.5 \div 2 = 0.75$ cm
Calculate the area:

$A = \pi(R^2 - r^2)$

$A = \pi(6^2 - 0.75^2)$

$A = \pi(36 - 0.5625)$

$A = 35.4375\pi$

$A = 111.33018... \text{ cm}^2$

Now use the volume formula for a prism: $V = Ah$
Substitute $A = 111.33018...$ and $h = 0.12$ :

$V = 111.33018... \times 0.12$

$V = 13.35962...$

Round to two decimal places:

$V \approx 13.36 \text{ cm}^3$

chatImportant

When you're given a diameter, always remember to divide by 2 to find the radius before using the formulas. This is one of the most common mistakes in volume calculations!

Understanding spheres and hemispheres

A sphere is a perfectly round three-dimensional object, like a ball. Every point on the surface of a sphere is the same distance from its centre. This distance is called the radius $(r)$ .

A hemisphere is exactly half of a sphere. You can imagine it as a ball cut in half through its centre.

Volume formulas for spheres and hemispheres

The volume of a sphere is calculated using the formula:

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

This means to find the volume, you cube the radius (multiply it by itself three times), multiply by $\pi$ , then multiply by $\frac{4}{3}$ .

Since a hemisphere is exactly half of a sphere, its volume is:

$V = \frac{1}{2} \times \frac{4}{3}\pi r^3 = \frac{2}{3}\pi r^3$

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Notice that half of $\frac{4}{3}$ equals $\frac{2}{3}$ . This makes the hemisphere formula easier to remember - you just need to change the numerator from 4 to 2!

Worked example: Finding the volume of a sphere

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Worked Example: Sphere Volume

Question: Find the volume of a sphere with radius $7$ m. Answer correct to one decimal place.

Solution:

Begin with the volume formula for a sphere:

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

Substitute $r = 7$ into the formula:

$V = \frac{4}{3} \times \pi \times 7^3$

Calculate the value:

$V = \frac{4}{3} \times \pi \times 343$

$V = 1436.75504...$

Round to one decimal place and include units:

$V \approx 1436.8 \text{ m}^3$

Worked example: Finding the volume of a hemisphere

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Worked Example: Hemisphere Volume

Question: Find the volume of a hemisphere with diameter $3.5$ m. Answer correct to one decimal place.

Solution:

Begin with the volume formula for a hemisphere:

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

Find the radius from the diameter. Since diameter $= 3.5$ m:

$r = 3.5 \div 2 = 1.75 \text{ m}$

Substitute $r = 1.75$ into the formula:

$V = \frac{2}{3} \times \pi \times 1.75^3$

Calculate the value:

$V = \frac{2}{3} \times \pi \times 5.359375$

$V = 11.2246...$

Round to one decimal place and include units:

$V \approx 11.2 \text{ m}^3$

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Key point: When you're given a diameter for a sphere or hemisphere, always divide by 2 to find the radius before using the volume formula. Many calculation errors occur from forgetting this crucial step!

Remember!

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Key Points to Remember:

Cylinder volume: $V = \pi r^2 h$ - multiply the circular base area by the height
Sphere volume: $V = \frac{4}{3}\pi r^3$ - cube the radius and multiply by $\frac{4}{3}\pi$
Hemisphere volume: $V = \frac{2}{3}\pi r^3$ - exactly half the volume of a sphere
Always convert diameter to radius by dividing by $2$ before using the formulas
Check your units carefully - volume is always measured in cubic units (mm³, cm³, m³)

Volume of Cylinders and Spheres (HSC SSCE Mathematics Standard): Revision Notes