### Video Transcript

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 ๐ฅ because theyโre 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 ๐ฟ, 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, ๐ฟ.

The lateral surface area is three hundred and ninety, so we can plug that in for ๐ฟ. We can bring down the equal sign and the one-half. Now for the perimeter ๐ โ thatโs the perimeter of the base โ and since we donโt know the side length of the base, weโre gonna have to leave ๐. And finally, our slant height ๐ is thirteen centimetres. Weโre 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 ๐ฅ.

So first, letโs take thirteen times one-half, which is six point five. Now to solve for ๐, 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 ๐ฅ plus ๐ฅ plus ๐ฅ plus ๐ฅ, it would equal sixty.

And ๐ฅ plus ๐ฅ plus ๐ฅ plus ๐ฅ is four ๐ฅ. Now to solve for ๐ฅ, we need to divide both sides by four, which means we would get ๐ฅ equals fifteen. Therefore, each side of the base is fifteen centimetres.