Find, to the nearest tenth, the surface area of this prism.
Now we’ve been given a triangular prism with a right triangle that’s the same shape all the way through the prism. It has a height of six millimeters, a depth of eight millimeters, and a length of 12 millimeters. Now the surface area of this prism is gonna be the sum of the areas of all the sides. Now if we look from this side, we can see we’ve got a right triangle with a base of 12 millimeters and a height of six millimeters. But because this is a prism, if I look from over here, I’m also gonna see the same sort of triangle. So there are gonna be two triangles like this contributing to the total surface area.
Then if we look at the base of this prism, I’ve got a rectangle with a length of 12 millimeters and a height of eight millimeters. Then looking at this end, I’ve got another rectangle, this time with a base length of eight millimeters and a height of six millimeters. And finally if I look directly down on the slope here, we know that it’s got a height of eight millimeters but we don’t actually know the length of this side. But of course we do know that it’s the hypotenuse of the triangle at the side, in fact the right triangle at the side, that we looked at first.
So let’s give that length a name; let’s call it 𝑥 millimeters. And because it’s a right triangle, we can use the Pythagorean theorem to work out the value of 𝑥. Now the Pythagorean theorem states that in a right triangle the square of the hypotenuse the longest side is equal to the sum of the squares of the other sides. So we know that 𝑥 squared is equal to six squared plus 12 squared. And six squared is 36, and 12 squared is 144. So 𝑥 squared is equal to 180. Taking square roots of both sides, we know that 𝑥 is equal to the square root 180, which simplifies to six times the square root of five. Now obviously we know that 𝑥 squared was equal to 180. So if 𝑥 would be negative six root five millimeters and we’d square that, we’d also have got 180. But because 𝑥 is length, we can ignore this negative value.
Now we know all our dimensions, we can go ahead work out the areas. To work out the area of the two right triangular sides, we know that the formula for the area of a triangle is a half times its base times its perpendicular height. So those two areas added together are two times a half times 12 times six. And that comes to 72 square millimeters. The base was a rectangle that had one side of 12 millimeters, the other of eight millimeters, and we can multiply those two together to find the area of the rectangle. And 12 times eight is 96 square millimeters. The area of the upright end of our shape was another rectangle with width eight millimeters and height six millimeters. And eight times six is 48 square millimeters. Then the area of the slope is six root five times eight, which is 48 root five square millimeters.
When I add those all together on my calculator, I get 323.3312629 and so on square millimeters. But the question asked us to round this to the nearest tenth. So looking at the tenth digit, that’s a three, we look at the hundredth digit just the right of it. That’s only a three; that’s less than five, so we’re not going around the first three up to a four. That’s gonna stay at 323.3. And of course, don’t forget the units: square millimeters.