### Video Transcript

π΄π΅πΆπ·π is a regular solid pyramid. π΄π΅πΆπ· is a square, where π΅πΆ is equal to eight centimetres. The angle between any sloping edge and the base π΄π΅πΆπ· is 65 degrees. Work out the surface area of the pyramid, giving your answer to two significant
figures.

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 π be the midpoint of the square π΄π΅πΆπ·, which forms the base of the
pyramid. And we shall let π be the midpoint of the line π΅πΆ.

Next, when we examine this question, itβs really important to pay attention to the
wording. The question tells us that the angle between any sloping edge and the base π΄π΅πΆπ·
is equal to 65 degrees. Now, it may be tempting to immediately draw on the right-angled triangle, here
represented as πππ.

We would then assume that the question is telling us that the angle marked πππ
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 π΄, π΅, πΆ, or π· to the top of the pyramid π. Here, we have marked on the sloping edge π΅π.

Now that we have done this, we can draw on a different right-angled triangle
ππ΅π. And we are now correct in saying that the angle marked here ππ΅π 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 πππ΅ since
the question has neither given us the side length ππ which is the height of the
pyramid or the side length π΅π.

Now, the solution to this problem has a lot of different components. So weβre gonna briefly spell out the different steps will be taken before
continuing.

The first thing weβre going to do is to find length π΅π using a different
right-angled triangle, which is triangle πππ΅. 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, weβll use triangle πππ΅ to find the length ππ or the height of the
pyramid. We will be able to do this using the newly found length π΅π and the angle of 65
degrees given in the question.

As step three, weβll find length ππ, using the triangle πππ 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 ππ΅πΆ on the diagram.

Finally, weβll 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 π΅π using triangle πππ΅.

First, we remember that we defined point π as the midpoint between π΅ and πΆ. The question gives us that the length π΅πΆ is equal to eight centimetres. And we can, therefore, say that π΅π is half of this length or four centimetres. We should also be able to see that the line of ππ 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.

Weβre therefore able to say that both π΅π and ππ are equal to four
centimetres. Now, it should be easy to see that triangle πππ΅ is a right-angled triangle. And because of this, we are able to use Pythagorasβs theorem to find the length
π΅π.

Pythagorasβs theorem tells us that the square of the hypotenuse, which is π΅π, is
equal to the sum of the square of the other two sides, π΅π squared plus ππ
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 π΅π squared is equal to
32. By taking the square root of both sides, we see that π΅π is equal to the square root
of 32. And we perform a simplification to find that π΅π is equal to four times root two
centimetres. Of course, since π΅π is a length, we ignore any negative solutions to the square
roots in our answers.

Great! We have now found π΅π using the triangle πππ΅. We now move on to finding the length ππ, which is the height of the pyramid. And we do so using triangle πππ΅. We can label the side length that we have just found π΅π 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
ππ΅ 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 π is equal to the opposite side divided by
the adjacent side. Letβs use this identity on our triangle.

We have that tan of 65 degrees is equal to the opposite side or ππ divided by the
adjacent side, which we know is four times the square root of two. To find ππ on its own, we can multiply both sides of our equation by four times the
square root of two and we find that ππ 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 ππ 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 youβre expecting. However, since weβll be using ππ at later stages in our calculations, but here
weβre going to choose not to approximate and to keep using the exact value for
ππ. We do this since weβll be using ππ in later calculations. And this helps preserve a higher degree of accuracy.

We have now found the value of ππ. And we can move on to step three, which is finding ππ using triangle πππ. For triangle πππ, we can label the side ππ, which is four centimetres, as shown
earlier. And we can label the side ππ, which we have just found of four times the square
root of two times tan 65 degrees in centimetres.

Weβre now in the situation again where we can use Pythagorasβs theorem to find the
desired length, which is the hypotenuse of this right-angled triangle ππ. Here, we have that the hypotenuse squared is equal to the sum of the square of the
other two sides. So ππ squared is equal to ππ squared plus ππ squared.

We can substitute in four for ππ and four times the square root of two times tan of
65 degrees for ππ. Taking the square root of both sides, we find that ππ 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 letβs find a value for ππ. Typing this into our calculator, we find that side ππ is equal to 12.774
centimetres to three decimal places.

Now that weβve found side length ππ, we can move on to our next step. For this step, we want to find the area of the triangle ππ΅πΆ, 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 π as the midpoint of the line π΅πΆ, we also know the height of
this triangle, which is side length ππ 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 π΅πΆ times ππ. Here, we substitute in our values to find half times eight times 12.774 is the area
of ππ΅πΆ. Performing this calculation, we find that we have a value of 51.094 centimetres
squared.

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 π΄π΅πΆπ·. The other four faces are triangles. These four slanted triangular faces are all identical to the triangle ππ΅πΆ, for
which we have just found the area.

The final and simple piece of the puzzle is, therefore, the area of the square
π΄π΅πΆπ·. The area of a square is equal to its side lengths squared. We can, therefore, say the area of π΄π΅πΆπ· 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 pyramid.

The total surface area is equal to the area of the square π΄π΅πΆπ· plus four times
the area of the triangle ππ΅πΆ. Weβre 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.