A square pyramid has a lateral surface area of three hundred and ninety
centimetres squared and a slant height of thirteen centimetres. Determine the length of each
side of its base.
We can go ahead and label the slant height of the square pyramid as thirteen
centimetres. We will also be finding the length of each side of its base. We can call them all
𝑥 because they’re all equal centi to square. Since we are given the lateral surface area to be three hundred and ninety
centimetres squared, we need to use the lateral surface area formula for a pyramid. It is 𝐿, which
is the lateral surface area of a regular pyramid, is equal to one-half of the perimeter of the
bases times the slant height, 𝐿.
The lateral surface area is three hundred and ninety, so we can plug that in for
𝐿. We can bring down the equal sign and the one-half. Now for the perimeter 𝑃 — that’s
the perimeter of the base — and since we don’t know the side length of the base, we’re gonna have
to leave 𝑃. And finally, our slant height 𝑙 is thirteen centimetres. We’re trying to find the length of each side of the base. So if we know the
perimeter of the base, which is the distance around that square, we can take the perimeter and
divide by four because the four numbers that would add up to our perimeter should all be equal
and that allows us to find 𝑥.
So first, let’s take thirteen times one-half, which is six point five. Now to solve for 𝑃, we divide both sides by six point
five, resulting in the perimeter of the base to be sixty. This means if you would add up all of the sides of the square, so 𝑥 plus 𝑥 plus
𝑥 plus 𝑥, it would equal sixty.
And 𝑥 plus 𝑥 plus 𝑥 plus 𝑥 is four 𝑥. Now to solve for 𝑥, we need to divide
both sides by four, which means we would get 𝑥 equals fifteen. Therefore, each side of the base is fifteen centimetres.