Video Transcript
This shape is composed of a rectangular prism and a right square pyramid. Calculate the surface area of the shape. If necessary, round your answer to two decimal places.
In this question, we’re asked to find the surface area of the shape. To find the surface area of a three-dimensional shape, we find the area of each individual face and then add all those areas together. So, let’s start by finding the area of one of the triangles in our pyramid. And to do this, we’re going to use the formula area of a triangle equals half times base length times the height.
Since this triangle is part of a square pyramid, we know that the base length will be four. However, we’re not given the height of this triangle. The value of three centimeters here refers to the height of the pyramid and not the triangle. In order to find the value of the height of the triangle, we’re going to need to use another triangle to help us.
So, let’s consider this internal triangle. If we call the height of our original triangle ℎ, this will correspond to the hypotenuse of this new triangle. The base length of our triangle will be two centimeters since it’s half of the length of our square. And the height of the triangle will be three centimeters since it’s the same as the height of the pyramid. Since we have a right triangle, we can use the Pythagorean theorem. This theorem says that the square of the hypotenuse is equal to the sum of the squares on the other two sides.
So, substituting in our values will give us ℎ squared equals three squared plus two squared. And it doesn’t matter which way round we have our three squared and our two squared. Since three squared is the same as three times three, which is nine, and two squared is the same as two times two, which is four, we can simplify this as ℎ squared equals nine plus four. And that gives us ℎ squared equals 13. In order to find the value of ℎ, we need to take the square root of both sides of the equation. This will give us ℎ equals root 13.
It’s always best to leave your answer in a form like root 13 if you can. But if you do want to change it into a decimal value, then an equivalent decimal would be 3.60555 and so on. And since our ℎ value is a length, the units for these values will be centimeters. And now, we’ve found the height of our triangle. Let’s go back to the original triangle and find the area.
Substituting our values four for the base length and root 13 for the height into the area formula will give us half times four times root 13. And since we know that half times four is equal to two, this will be two times root 13, which we can write as two root 13 centimeters squared. So, now, we’ve found the area of one triangle, we can work out the surface area of the pyramid by multiplying the triangle area by four since there are four triangular faces.
So, the surface area of the pyramid is equal to four times two root 13. And since four times two is eight, we can write this as eight root 13 centimeters squared. The equivalent decimal here would be 28.844 recurring centimeters squared. So, now, we’ve found the surface area of the pyramid, let’s move on to finding the area of one of the rectangles on the side. And to do this, we’re going to use the formula the area of a rectangle equals length times width.
So, substituting in our values four and nine centimeters will give us the area of a rectangle equals four times nine, which we can evaluate as 36 centimeters squared. And now since we’ve found the area of one rectangle, we can use this to find the surface area of all the vertical sides. Since we know that we have four identical rectangles, we can write the surface area of the vertical sides as four times 36. And evaluating this will give us an answer of 144 centimeters squared.
And now, there’s one more face to find. Can you see what it is? It’s the area of the square on the base of our rectangular prism. Since the area of a square is calculated by the length times the length, we can substitute our length value to give us four times four, giving us an area of 16 centimeters squared. So, finally then, to put it all together and calculate the total surface area, we add together the surface area of our pyramid, the surface area of the vertical sides, and the area of the square at the base of our rectangular prism.
This will give us a surface area of eight root 13 plus 144 plus 16. And we use our calculator to evaluate eight root 13, which will give us an answer of 188.8444 and so on. However, since we’re asked for an answer to two decimal places, we need to check our third decimal place digit to see if it’s equal to five or more. In this case, our value is four, so we don’t round up. Meaning our final answer for the surface area is 188.84 centimeters squared.
