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Stage 5.1-3

7.04 Volume of prisms and cylinders

Lesson

Volume of prisms

We can find the volume of any prism by multiplying the base area by the height of the prism, where the height of the prism is the distance between the two base faces.

The volume of a prism is given by V=Ah where A is the base area and h is the height of the prism.

Exploration

Use the applet below to explore the volume of prisms with triangular and rectangular bases.

Loading interactive...

For both prisms the volume is found by multiplying the area of the base by the height of the prism.

Examples

Example 1

Find the volume of the triangular prism shown.

Triangular prism with base length of 5 centimetres, base height of 5 centimetres, and prism height of 10 centimetres.
Worked Solution
Create a strategy

Use the volume formula V=Ah, where the area of the base is given by A=\dfrac{1}{2} b H.

Apply the idea

We are given base length b=5, height of the triangular base H=5, and height of prism h=10.

\displaystyle V\displaystyle =\displaystyle \left(\dfrac{1}{2} b H\right) hWrite the formula
\displaystyle =\displaystyle \left(\dfrac{1}{2} \times 5 \times 5 \right)\times 10Subsitute the values
\displaystyle =\displaystyle 125 \text{ cm}^3Evaluate

Example 2

A prism has a volume of 1080 \text{ cm}^3.

If it has a base area of 120 \text{ cm}^2, find the height of the prism.

Worked Solution
Create a strategy

Use the volume of a prism formula: V=A h.

Apply the idea

We are given V=1080 and A=120.

\displaystyle V\displaystyle =\displaystyle A hWrite the formula
\displaystyle 1080\displaystyle =\displaystyle 120\times hSubsitute the values
\displaystyle \dfrac{1080}{120}\displaystyle =\displaystyle \dfrac{120\times h}{120}Divide both sides by 120
\displaystyle h\displaystyle =\displaystyle 9 \text{ cm}Evaluate
Idea summary

The volume of a prism is given by:

\displaystyle V=Ah
\bm{A}
is the base area
\bm{h}
is the height of the prism

Volume of cylinders

We can find the volume of a cylinder using the same method, by multiplying the area of the circular base by the height of the cylinder.

Since the area of a circle is given by the formula A=\pi r^2, the formula for the volume of a cylinder is: V=\pi r^2h, where r is the radius and h is the height of the cylinder.

Exploration

Use the applet below to explore the volume of cylinders.

Loading interactive...

As the height and radius of a cylinder increase so does its volume.

Examples

Example 3

Calculate the volume of the solid. Assume that the solid is a quarter of a cylinder.

Round your answer to one decimal place.

A quarter of cylinder with radius of 5 centimetres and height of 14 centimetres.
Worked Solution
Create a strategy

Multiply the volume of a cylinder formula by a quarter.

Apply the idea

We are given r=5 and h=14.

\displaystyle V\displaystyle =\displaystyle \dfrac{1}{4}\pi r^2 hMultiply the formula by \dfrac{1}{4}
\displaystyle =\displaystyle \dfrac{1}{4}\pi \times 5^2 \times 14Subsitute the values
\displaystyle =\displaystyle 274.9 \text{ cm}^3Evaluate

Example 4

A wedding cake with three tiers rests on a table. The layers have radii of 50 cm, 54 cm and 58 cm as shown in the figure. If each layer is 21 cm high, calculate the total volume of the cake in cubic metres. Round your answer to two decimal places.

This image shows 3 cylinders on top of each other, all with a height of 21 centimetres. Ask your teacher for more information
Worked Solution
Create a strategy

Add the three volumes and convert the units to cubic metres.

Apply the idea
\displaystyle \text{Total volume}\displaystyle =\displaystyle \pi r_1^2 h +\pi r_2^2 h +\pi r_3^2 hAdd the volumes
\displaystyle =\displaystyle \pi \times 50^2 \times 21+\pi \times 54^2 \times 21+\pi \times 58^2 \times 21Substitute the radii and heights
\displaystyle \approx\displaystyle 579\,247\text{ cm}^3Evaluate

To convert to metres cubed we need to divide by 100^3.

\displaystyle V\displaystyle =\displaystyle \dfrac{579\,247}{1\,000\,000}Convert to \text{m}^3
\displaystyle =\displaystyle 0.58 \text{ m}^3Evaluate and round
Idea summary

The volume of a cylinder is given by:

\displaystyle V = \pi r^2 h
\bm{r}
is the radius of the cylinder
\bm{h}
is the height of the cylinder

Outcomes

MA5.2-11MG

calculates the surface areas of right prisms, cylinders and related composite solids

MA5.2-12MG

applies formulas to calculate the volumes of composite solids composed of right prisms and cylinders

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