Question

What is the total area of the square pyramid below?

Answer 1

That pyramid is SERIOUSLY "not to scale"!! :-)

It looks like the length of MV is 3, yes?

Okay. So since all the faces are the same, we only need to find the area of one.

To do that, we need to know the base of a face and the "slant height".

And the "slant height" happens to be the hypotenuse of the triangle VMP.

Do you see that?

Answer 2

yes, would that always be the slant height?

Answer 3

24

Answer 4

On a pyramid, yes.

Most triangles are staring at you straight on, so they have no slant. So their vertical height is the same as the line that goes up their center.

If you tilt a triangle, then the slant height remains the same while the vertical height gets smaller.

Answer 5

@armn, way to show HOW to do the problem.

This forum isn't for showing everybody how breathtakingly brilliant YOU are, it's to TEACH people how to do problems!

Answer 6

area = 1/2*base area*height

Answer 7

Yeah, well she first has to know how to find that height.

Answer 8

3 is the height....its in the fig.

Answer 9

No, 3 is the height of the pyramid. It is NOT the height of a face...look at the figure again.

Answer 10

uh.....dont u want the total height of the pyramid?

Answer 11

*area not height srry

Answer 12

@lana.831 are you still there?

Answer 13

@qweqwe123123123123111 i had to take care of something. I'm off and on this website all day sorry about that. but yes, i see what you're saying.

Answer 14

Okay...so do you see where we got the "slant height" from? And can you figure out what that height is?

Answer 15

yeah. so would the height be 3 too?

Answer 16

No...

Look at the diagram.

Do you see the triangle VMP?

The edge VM is 3, and is the vertical height of the pyramid.
MP is the base of that triangle, and is half of 4.

The SLANT HEIGHT of a triangular face on the pyramid is the hypotenuse of VMP, which is line VP

Do you see that?

Answer 17

why half of 4?

Answer 18

I see that it's VP but how is that the hypotenuse of VMP?

Answer 19

Because each side of the base of the pyramid is 4.

The line MP is drawn from the CENTER of that square, which is half way across the base.

Half way across 4 is 2.

Answer 20

ok i get that now..

Answer 21

-"I see that it's VP but how is that the hypotenuse of VMP"

Because VM and MP form a right triangle. VP is the hypotenuse of that triangle

Answer 22

So...can you tell me the length of VP? :-)

Answer 23

Pythagoras sez that the hypotenuse^2 is equal to the sum of the sides squared, like this:.

VP^2 = VM^2 + MP^2

Which means that:

VP = sqrt(VM^2 + MP^2)

Answer 24

the \[\sqrt{13}\] ?

Answer 25

YES!!!! Excellent!!! :-)

You now have the "slant height" of a face of the pyramid!

Can you now tell me the area of the face?

Answer 26

Area of triangle:

1/2 B*H

In this case, the "H" is the "slant height" we just came up with.

Answer 27

3.6?

Answer 28

Yup!!

3.60555 (to 5 places)

That's the "slant height" of a face.

Answer 29

BTW, I just get tediously accurate sometimes. "3.6" is probably sufficient. :-)

Answer 30

lol it's all good. so where do we go from here?

Answer 31

Now that we have the slant height of a face, we can use that to find the area of that face.

Which as I mentioned above:

Area of triangle:

1/2 B*H

In this case, the "H" is the "slant height" we just came up with.