Find the lateral area of the given prism to the nearest square centimeter.
Lateral area is the surface area of the sides excluding the top and bottom. So here it’s not the top and bottom because the bottom actually isn’t the base. It’s not laying on the bottom. This is a triangular prism.
A prism is made up of rectangles and its two bases. And the bases are what distinguish what kind of prism it is. So here we have the triangles as our bases. So it becomes a triangular prism. So the lateral area will be the area of the sides excluding the top and bottom, which are the bases, the triangles.
So the area that we need to find will be this rectangle, which is a 10 by 16, because we know this length is 16. We also need to find the area of this rectangle. And that’s a 16 by — we actually don’t know that length. But we do know we have a right triangle with sides 10 and 15. So we can use the Pythagorean theorem to find it.
10 and 15 would be the legs. And we can call the hypotenuse 𝑥, because the Pythagorean theorem states the square of the longest side, the one across from the 90-degree angle, is equal to the sum of the squares of the shorter sides, the 10 and 15. So 100 plus 225 is 325.
And now we need to square-root both sides, which is about 18.03. So we can go ahead and label that on our diagram. So here we’ve recognized the two rectangles we need to find the area for to find our lateral area. There’s one more rectangle, the one on the bottom. And it’s 15 by 16. So let’s write out all of the areas that we need to find.
The first rectangle was 10 by 16. And we find the area of a rectangle by length times width, so 10 times 16. Our next rectangle is 16 by 18.03. So we’ll multiply those. And lastly, we have a 15-by-16 rectangle. So we need to multiply and then add these together. So we have 160 plus 288.48 plus 240, which gives us 688.48.
However, it says to round to the nearest square centimeter. So we look here at the four. Since the four is less than five, it will keep this eight an eight, resulting in 688 centimeters squared because this is an area. So it should be square centimeters.
Now there’s also another way to do this problem. We could have used the formula for lateral area, which is the perimeter of the base times the height of the prism itself. So the bases were the triangles. So in order to find the perimeter, we need to add up all of the sides. So we have 10 plus 15 plus 18.03, giving us 43.03.
Now the height, the height of a prism is the distance between the bases. So here’s our other triangle, the other base. So the distance between these two triangles would be 16. So we have 43.03 times 16, which gives us 688.48, which is exactly what we got before. So we would have rounded to be 688 square centimeters. So either way will work. And we would still result in the same answer, 688 square centimeters.