HOW DO YOU FIND THE TOTAL SURFACE AREA OF A SQUARE-BASED PYRAMID?
Find the slant height first, from the right triangle whose legs are the perpendicular height and half the base. The perpendicular height is not the slant height, and using it as one is the single most common way to lose this question.
A right pyramid has a square base of side 6 cm and a perpendicular height of 4 cm. What is its total surface area, in cm2?
The slant height is not given and is not the same as the height. Get it from the right triangle on the apothem first, then everything else follows.
- apothem = 6 ÷ 2 = 3
- l = √(32 + 42) = √25 = 5
- 4 triangular faces = 4 × ½ × 6 × 5 = 60
- square base = 6 × 6 = 36
- total = 60 + 36 = 96 cm2
84 is what you get if you use 4 as the slant height instead of the perpendicular height. The 4 is the height of the apex above the centre of the base; the slant height runs up the middle of a triangular face and is always longer. It is the single most common way to lose this question.
60 is what you get if you forget the base and total only the four triangular faces. "Total surface area" includes the base unless the question says the solid is open.
3, 4, 5 is the first triple to test whenever a pyramid gives you an even base and a whole height. You should not be reaching for Pythagoras here.
THE GENERAL RULE
Any question about the surface of a pyramid or a cone is really a question about which right triangle you build first. Draw the one standing on the apothem — half the base along the bottom, the perpendicular height up the middle — and the slant height falls out as its hypotenuse. Reading the perpendicular height straight into the face-area formula is the error every one of these questions is written to catch.
Updated · written to the published ESAT content specification