A regular square pyramid has base edges 10 in. long and height 4 in. Find the total surface area. Round your answer to the nearest tenth.

Answer 1

Hello Audrey!

I'm hoping saying this in text will make some sense.

Imagine you're standing on one side of the pyramids of Giza. In your perspective, you will just see a humongous slanted triangle. But, if you imagine the cross section if you cut the pyramid into 4 equal pieces heightwise, you're actually just viewing the a triangle's hypotenuse.

Given that the pyramid's height is 4in long, that is just the opposite side of that same cross section triangle.

Also, given that each edge is 10in long, then the distance between the edge to the very center (where you'll find the base of the opposite side), which is 5in since the base is just a regular square (All base edges are equal).

Then...

|\

| \

4in -> | \

| \

| \

---------- o <- you are here (facing towards the pyramid face)

5in

Now, if we found the missing length (the hypotenuse length being the slant you're facing), then we can the area of that pyramid face and by finding the area of one face, we can find the area of all 4 pyramid faces since it's structured on a regular square from the premise.

Using pythagorean theorm (a^2 + b^2 = c^2) since the triangle above is very much a 90 degree triangle where 'c' is the length of the hypotenuse and 'a' and 'b' are the sides with lengths i described earlier...

(5)^2+(4)^2 = c^2

(25) + (16) = c^2

sqrt(41) = c (We take the positive 'c' after taking sqrt of both sidesbecause there is no such thing as negative length)

Now our hypotenuse is c = sqrt(41).

Moving forward, recall that the area of a right triangle is (1/2)*(base)*(height), but upon inspection of the triangle face you're looking at and not the cross section, it should resemble an isosceles triangle, which means the area should just be (base)*(height).

Do be careful with 'height' though, as I am not talking about the height given to us (4in). The hypotenuse we just solved for is the height of the slanted triangle you are observing and the height we will be using to find the area of the slanted triangle and the base is 10in. So, (base)*(height) = (sqrt(41))*(10) and recall that it's just for a single slanted triangle face.

We now just simply multiply that result by 4 since there are 4 slanted triangle faces, then add it to the area of the base since the bottom of the pyramid is very much still a qualified surface. The area of just the base is (10*10) since the area of a square in this context is (edge*edge).

Now...

4*(10*sqrt(41)) + (10*10) = (total area of each slanted triangle face) + (Area of the base)

= 40*sqrt(41) + 100

Hope this helps!