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