A square pyramid has a lateral
surface area of 42 square yards. If its slant height is three yards,
determine its total surface area.
Let’s start by sketching out the
square pyramid and filling in the relevant information. We’re told that this square pyramid
has a lateral surface area of 42 square yards. The lateral surface area of a
pyramid is the total area of the lateral sides or the triangles, but not including
the area of the base. So the lateral surface area of this
square pyramid will be the area of the triangle at the back plus the area of the two
triangles at the sides plus the area of the triangle at the front. As we’re told that the lateral
surface area is equal to 42 square yards, this is equivalent to saying that four
times the area of one of these triangles is equal to 42. And therefore, the area of one
triangle is equal to 42 divided by four or 10.5 square yards.
We can then take this information
about the area of the triangle and combine it with the perpendicular height in order
to work out the base length of this triangle. Knowing this would then allow us to
calculate the area of the square on the base of this pyramid. We can recall that the area of a
triangle is equal to a half multiplied by the base multiplied by the height. For our triangle then, we have an
area of 10.5 and an unknown base length of 𝑏 and a height of three. Simplifying, we have that 10.5 is
equal to three-halves 𝑏. In the next step of rearranging, we
divide by three-halves, which is equivalent to multiplying by two-thirds. And so we’ve found that the base
length 𝑏 of the triangle is equal to seven yards.
As the base of the triangle is also
equivalent to the length of the square, we know that all the lengths on the square
will be seven yards. To find the area of a square, we
multiply the length by the length, which is seven times seven, giving us a value of
49 square yards. Finally, then, to find the total
surface area, we were told that the lateral surface area is 42 square yards. That’s the area of all four
triangles at the top of the pyramid. Therefore, for the total surface
area, we must add on the area of the square to this lateral surface area, giving us
a value of 91 square yards for the total surface area.