Pyramids

Calculate the volume of a square-based pyramid with a base side length of $14$ cm and a perpendicular height of $9$ cm.

Step 1: Calculate the base area.

• Base area = $14 \times 14 = 196$ cm²

Step 2: Substitute the values into the volume formula.

• $V = \frac{1}{3} \times 196 \times 9$

Step 3: Calculate the final answer with units.

• $V = 588$ cm³

Surface Area of a Pyramid

Wrapping a pyramid-shaped gift requires you to figure out exactly how much paper will cover the outside. The total surface area is simply the sum of the areas of all the individual 2D shapes that make up the solid.

For a pyramid, this means adding the area of the base to the lateral surface area (the total area of the triangular faces). Unlike volume, the surface area formula is not given in your exam.

$$\text{Total Surface Area} = \text{base area} + \text{sum of triangular face areas}$$

To find the area of the triangular faces, you must use the slant height. This is the distance from the apex down the middle of a slanted face to the edge of the base. It acts as the 2D height for the individual triangles. Surface area is always measured in square units, such as cm².

Worked Example: Calculating Surface Area

Calculate the total surface area of a regular square-based pyramid with a base side length of $8$ cm and a slant height of $11$ cm.

Step 1: Calculate the base area.

• Base area = $8 \times 8 = 64$ cm²

Step 2: Calculate the area of one triangular face.

• Area of one triangle = $\frac{1}{2} \times \text{base} \times \text{height}$
• Area = $\frac{1}{2} \times 8 \times 11 = 44$ cm²

Step 3: Calculate the sum of all triangular face areas.

• A square-based pyramid has $4$ identical triangular faces.
• $4 \times 44 = 176$ cm²

Step 4: Add the base area to find the total surface area.

• Total Surface Area = $64 + 176 = 240$ cm²

Using 3D Pythagoras for Missing Heights

You cannot always rely on the examiner to hand you the exact height you need for a formula. Higher Tier questions often give you the slant height when you actually need the perpendicular height for volume, or vice versa.

You can find the missing value by creating a right-angled triangle inside the pyramid. This internal triangle is formed by the perpendicular height, the slant height (which acts as the hypotenuse), and half the width of the base. By applying Pythagoras' Theorem ($a^2 + b^2 = c^2$), you can calculate the required height before moving on to the volume or surface area formulas.