Volume of a Pyramid Revision Notes for VCE SSCE General Mathematics

Volume of a Pyramid

Understanding pyramids and prisms

A pyramid is a three-dimensional shape with a polygon base and triangular faces that meet at a single point called the apex. When you place a pyramid inside a prism that has the same base and height, you'll discover an important relationship.

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The pyramid takes up exactly one-third of the space inside the prism. This one-third relationship is fundamental to calculating the volume of any pyramid.

Volume formula for pyramids

Because a pyramid occupies one-third of the volume of its corresponding prism, we can write:

$\text{Volume of pyramid} = \frac{1}{3} \times \text{volume of prism}$

Since the volume of a prism equals the area of its base multiplied by its height, this becomes:

$V = \frac{1}{3} \times \text{area of base} \times \text{height}$

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This formula works for any pyramid, regardless of the shape of its base - whether it's square, rectangular, triangular, hexagonal, or any other polygon.

Special case: square and rectangular pyramids

For pyramids with square or rectangular bases, we can write the formula in terms of the base dimensions:

$V = \frac{1}{3}lwh$

where:

$l$ = length of the base
$w$ = width of the base
$h$ = perpendicular height of the pyramid

chatImportant

The height must always be the perpendicular (vertical) distance from the base to the apex, not the slant height along the face of the pyramid. Using the slant height is a common mistake that will give you an incorrect answer.

Worked example: square pyramid

Let's find the volume of a square pyramid with a height of $11.2$ cm and a base length of $17.5$ cm. We need to give our answer to two decimal places.

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Worked Example: Finding the Volume of a Square Pyramid

Step 1: Write down the formula

$V = \frac{1}{3} \times \text{area of base} \times \text{height}$

Step 2: Identify the values

The base is a square with side length $17.5$ cm, so:

Area of base = $17.5^2$ cm²
Height = $11.2$ cm

Step 3: Substitute and calculate

\begin{aligned} V &= \frac{1}{3} \times 17.5^2 \times 11.2 \\ &= \frac{1}{3} \times 306.25 \times 11.2 \\ &= \frac{1}{3} \times 3430 \\ &= 1143.333... \end{aligned}

Step 4: Round and include units

The volume of the pyramid is 1143.33 cm³ (to 2 decimal places).

Worked example: hexagonal pyramid

Now let's find the volume of a hexagonal pyramid that has a base area of $122$ cm² and a height of $12$ cm.

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Worked Example: Finding the Volume of a Hexagonal Pyramid

Step 1: Write down the formula

$V = \frac{1}{3} \times \text{area of base} \times \text{height}$

Step 2: Identify the values

We're given:

Area of base = $122$ cm²
Height = $12$ cm

Notice that we don't need to calculate the area of the hexagonal base because it's already given.

Step 3: Substitute and calculate

\begin{aligned} V &= \frac{1}{3} \times 122 \times 12 \\ &= \frac{1}{3} \times 1464 \\ &= 488 \end{aligned}

Step 4: Include units

The volume is 488 cm³.

Exam tips

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Tips for Success:

Always check whether you're given the area of the base or the dimensions of the base. If you're given dimensions, you'll need to calculate the area first.
Make sure you're using the perpendicular height, not the slant height.
Remember to cube your units (cm³, m³, etc.) for volume.
Show your working clearly, including substitution into the formula.
Round your final answer appropriately as instructed in the question.

Remember!

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Key Points to Remember:

A pyramid occupies exactly one-third of the volume of a prism with the same base and height.
The general formula for any pyramid is: $V = \frac{1}{3} \times \text{area of base} \times \text{height}$
For square or rectangular pyramids: $V = \frac{1}{3}lwh$
Always use the perpendicular height (vertical distance from base to apex), not the slant height.
Don't forget to include cubic units (cm³, m³) in your final answer.

Volume of a Pyramid (VCE SSCE General Mathematics): Revision Notes