𝐴𝐵𝐶𝐷𝑃 is a regular solid pyramid. 𝐴𝐵𝐶𝐷 is a square, where 𝐵𝐶 is equal to eight centimetres. The angle between any sloping edge and the base 𝐴𝐵𝐶𝐷 is 65 degrees. Work out the surface area of the pyramid, giving your answer to two significant
Now, before we delve into this question, let us define two more points to make the
different components of our pyramid easier to refer to. We shall let 𝑌 be the midpoint of the square 𝐴𝐵𝐶𝐷, which forms the base of the
pyramid. And we shall let 𝑋 be the midpoint of the line 𝐵𝐶.
Next, when we examine this question, it’s really important to pay attention to the
wording. The question tells us that the angle between any sloping edge and the base 𝐴𝐵𝐶𝐷
is equal to 65 degrees. Now, it may be tempting to immediately draw on the right-angled triangle, here
represented as 𝑃𝑋𝑌.
We would then assume that the question is telling us that the angle marked 𝑃𝑋𝑌
here is equal to 65 degrees. However, doing this would actually be incorrect. To see why, let us recall that in a 3D shape, an edge is the line segment where two
faces meet. And the face is one of the flat surfaces on the solid object.
We should, therefore, be able to see that the sloping edges go from any of the four
corners of the square 𝐴, 𝐵, 𝐶, or 𝐷 to the top of the pyramid 𝑃. Here, we have marked on the sloping edge 𝐵𝑃.
Now that we have done this, we can draw on a different right-angled triangle
𝑃𝐵𝑌. And we are now correct in saying that the angle marked here 𝑃𝐵𝑌 is indeed equal to
65 degrees. And now that we understand what the question is telling us, we begin to wonder how
this helps us. Unfortunately, we do not have a lot of information about this triangle 𝑃𝑌𝐵 since
the question has neither given us the side length 𝑃𝑌 which is the height of the
pyramid or the side length 𝐵𝑌.
Now, the solution to this problem has a lot of different components. So we’re gonna briefly spell out the different steps will be taken before
The first thing we’re going to do is to find length 𝐵𝑌 using a different
right-angled triangle, which is triangle 𝑌𝑋𝐵. We should be able to do this using only the side length of the square, which is given
in the question as eight centimetres.
Next, we’ll use triangle 𝑃𝑌𝐵 to find the length 𝑃𝑌 or the height of the
pyramid. We will be able to do this using the newly found length 𝐵𝑌 and the angle of 65
degrees given in the question.
As step three, we’ll find length 𝑃𝑋, using the triangle 𝑃𝑌𝑋 and the information
that we have already found at this point. Once we have done this, we will know both the base and the height of the triangle
which forms the four slanted triangular faces of the pyramid. This information will allow us to calculate the area of these faces. Here, we have marked the area 𝑃𝐵𝐶 on the diagram.
Finally, we’ll add the area of our four triangular faces to the area of the square
base to find the total surface area of the pyramid.
Okay, now that we understand the process that we will be going through, let us get to
the solution. Step one, find 𝐵𝑌 using triangle 𝑌𝑋𝐵.
First, we remember that we defined point 𝑋 as the midpoint between 𝐵 and 𝐶. The question gives us that the length 𝐵𝐶 is equal to eight centimetres. And we can, therefore, say that 𝐵𝑋 is half of this length or four centimetres. We should also be able to see that the line of 𝑌𝑋 is also equal to four centimetres
since the distance from the midpoint of any of the sides of the square to the centre
of the square is equal to half of the side lengths.
We’re therefore able to say that both 𝐵𝑋 and 𝑌𝑋 are equal to four
centimetres. Now, it should be easy to see that triangle 𝑌𝑋𝐵 is a right-angled triangle. And because of this, we are able to use Pythagoras’s theorem to find the length
Pythagoras’s theorem tells us that the square of the hypotenuse, which is 𝐵𝑌, is
equal to the sum of the square of the other two sides, 𝐵𝑋 squared plus 𝑌𝑋
squared. We substitute in our value of four for both of these sides. We then square our fours to get 16 plus 16 and we find that 𝐵𝑌 squared is equal to
32. By taking the square root of both sides, we see that 𝐵𝑌 is equal to the square root
of 32. And we perform a simplification to find that 𝐵𝑌 is equal to four times root two
centimetres. Of course, since 𝐵𝑌 is a length, we ignore any negative solutions to the square
roots in our answers.
Great! We have now found 𝐵𝑌 using the triangle 𝑌𝑋𝐵. We now move on to finding the length 𝑃𝑌, which is the height of the pyramid. And we do so using triangle 𝑃𝑌𝐵. We can label the side length that we have just found 𝐵𝑌 which is four times the
square root of two centimetres. We can also draw on the angle that the question gives us between the sloping edge
𝑃𝐵 and the base which is 65 degrees.
This situation with a right-angled triangle should be familiar to us. We have an angle, unknown adjacent side, and an unknown opposite side that we want to
find. In this situation, we can use trigonometry to help us.
Looking at SOHCAHTOA, we can see that we need to use the tan function given by
TOA. The identity here is that tan of an angle 𝜃 is equal to the opposite side divided by
the adjacent side. Let’s use this identity on our triangle.
We have that tan of 65 degrees is equal to the opposite side or 𝑃𝑌 divided by the
adjacent side, which we know is four times the square root of two. To find 𝑃𝑌 on its own, we can multiply both sides of our equation by four times the
square root of two and we find that 𝑃𝑌 is equal to four times the square root of
two times tan of 65 degrees.
When working with trig functions, it is always worth checking that your calculator is
set to the right units. And in this case, we want degrees instead of radians. At this stage, we could type the calculation into our calculator to find the length
of 𝑃𝑌 is approximately equal to 12.131 centimetres.
This may be good as a sense check to make sure that the value is roughly within the
range that you’re expecting. However, since we’ll be using 𝑃𝑌 at later stages in our calculations, but here
we’re going to choose not to approximate and to keep using the exact value for
𝑃𝑌. We do this since we’ll be using 𝑃𝑌 in later calculations. And this helps preserve a higher degree of accuracy.
We have now found the value of 𝑃𝑌. And we can move on to step three, which is finding 𝑃𝑋 using triangle 𝑃𝑌𝑋. For triangle 𝑃𝑌𝑋, we can label the side 𝑋𝑌, which is four centimetres, as shown
earlier. And we can label the side 𝑃𝑌, which we have just found of four times the square
root of two times tan 65 degrees in centimetres.
We’re now in the situation again where we can use Pythagoras’s theorem to find the
desired length, which is the hypotenuse of this right-angled triangle 𝑃𝑋. Here, we have that the hypotenuse squared is equal to the sum of the square of the
other two sides. So 𝑃𝑋 squared is equal to 𝑋𝑌 squared plus 𝑃𝑌 squared.
We can substitute in four for 𝑋𝑌 and four times the square root of two times tan of
65 degrees for 𝑃𝑌. Taking the square root of both sides, we find that 𝑃𝑋 is equal to the square root
of 16 plus four times the square root of two times tan of 65 degrees squared. Although we want to keep our answers as accurate as possible, here our line of
working is getting a little bit big. So let’s find a value for 𝑃𝑋. Typing this into our calculator, we find that side 𝑃𝑋 is equal to 12.774
centimetres to three decimal places.
Now that we’ve found side length 𝑃𝑋, we can move on to our next step. For this step, we want to find the area of the triangle 𝑃𝐵𝐶, which is one of the
triangular slanted faces of the pyramid. We know the base of this triangle is eight centimetres as it is given in the
question. Since we define point 𝑋 as the midpoint of the line 𝐵𝐶, we also know the height of
this triangle, which is side length 𝑃𝑋 that we just found.
The area of a triangle we know is half times the base times the height. In our case, the area is half times 𝐵𝐶 times 𝑃𝑋. Here, we substitute in our values to find half times eight times 12.774 is the area
of 𝑃𝐵𝐶. Performing this calculation, we find that we have a value of 51.094 centimetres
We now come to our final stage where we calculate the surface area of the
pyramid. The surface of the pyramid consists of five different faces. One of these is the square which forms the base, which is the shape 𝐴𝐵𝐶𝐷. The other four faces are triangles. These four slanted triangular faces are all identical to the triangle 𝑃𝐵𝐶, for
which we have just found the area.
The final and simple piece of the puzzle is, therefore, the area of the square
𝐴𝐵𝐶𝐷. The area of a square is equal to its side lengths squared. We can, therefore, say the area of 𝐴𝐵𝐶𝐷 is equal to eight squared centimetres
squared or 64 centimetres squared. We are now ready to perform our final calculation and find the total surface area of
The total surface area is equal to the area of the square 𝐴𝐵𝐶𝐷 plus four times
the area of the triangle 𝑃𝐵𝐶. We’re able to substitute in the values we have found for the area of these
shapes. Performing this calculation, we find that the surface area of the pyramid is
approximately equal to 268.376 centimetres squared.
Finally, we recall that the question asked for our answer to two significant
figures. Rounding our answer, we say that the total surface area of the pyramid is equal to
270 centimetres squared to two significant figures.