Square pyramid

Properties

• 5 faces, 8 edges and 5 corners.
• The four side faces are equal isosceles triangles.
• The height runs from the top to the centre of the base. The slant height runs down the middle of a side face.

Why the formulas work

Any pyramid has one third of the volume of the prism with the same base and height, just as a cone is one third of its cylinder. Here the prism is a box of volume a²h, so the pyramid is ⅓a²h. Each side face is a triangle with base a and height l, so the four together have area 4 × ½al = 2al.

Worked example

Take a square pyramid with base side 10 cm and height 12 cm.

1. Slant height. Slant height l = √(h² + a² ⁄ 4). Here: √(12² + 10² ⁄ 4) = 13 cm.
2. Volume. Volume = ⅓ a²h. Here: ⅓ × 10 × 10 × 12 = 400 cm³.
3. Lateral surface area. Lateral surface area = 2al. Here: 2 × 10 × 13 = 260 cm².
4. Total surface area. Total surface area = a² + 2al. Here: 10² + 2 × 10 × 13 = 360 cm².

Finding a missing measurement

From the slant height, h = √(l² − a² ⁄ 4): a base side of 10 cm and a slant height of 13 cm give h = √(169 − 25) = 12 cm. From the volume, h = 3V ÷ a²: a volume of 400 cm³ on that base gives 1200 ÷ 100 = 12 cm.

Mistakes to avoid

• Using the vertical height for the area of the side faces. They need the slant height.
• Forgetting the one third in the volume.

Where you see it

The pyramids of Giza, some tents, the roofs of towers and gazebos, and certain paperweights are square pyramids.

Try these

Work each one out, then tap to check.

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