What is the surface area and volume of a pentagonal pyramid with a slant height of 12.5 and side length of 11?

Question

Please help! Explain :) Thank you!!

Answer 1

Explaining this would be a lot easier if I could draw a diagram, but I can't. I hope the following explanation will suffice.

 

If this was a regular pentagonal pyramid, the slant height (length from the apex to the middle of one of the triangular faces) would be equal to (√3/2) times the edge (11), which would be 9.526. Since the slant height is 12.5, this is not a regular pentagonal pyramid, meaning the triangular faces are not equilateral triangles, so we have to do this in pieces. Since the slant height is given, the surface area of the triangular faces is easy to find, with each triangular face having an area of (1/2)bh, or (1/2) (11)12.5 = 68.75. Since there are 5 triangular faces, the surface area of all the triangular faces together is 5(68.75) = 343.75.

 

Now the area of the base pentagon must be found from the edge length. Since the apothem (distance from the midpoint of the edge to the center of the pentagon) is not given, it must be calculated. If you divide the pentagon into 5 equal triangles from the center, the top angle of one of these triangles is 360/5 = 72 degrees. Dividing this isosceles triangle in half gives two right triangles. The angle at the top is 1/2 of 72° = 36°. The tangent of this angle can be expressed as side opposite divided by side adjacent, or half the edge (11/2) divided by the apothem (a).

 

tan 36° =   side opposite        =  5.5    

                side adjacent              a

 

a =   5.5         =  7.5701    The area of each of the 5 triangles making up the pentagon base is 

       tan 36°                          (1/2)ba  = (1/2)(11)(7.5701) = 41.6356

                                        The area of the pentagon base is therefore 5(41.6356) = 208.18

 

The surface area of the entire pentagonal pyramid is therefore 343.75 + 208.18 = 551.93

 

The volume of a pentagonal pyramid is (5/6) abh, where a is the apothem of the base, b is the length of the side of the base, and h is the height of the pyramid, which is the one value we do not yet have.

 

                        V = (5/6) abh

 

However, it can be calculated from the apothem and the slant height, because the height of the pyramid forms a right triangle with the apothem as the other side and the slant height as the hypoteneuse.  Therefore, the height of this pyramid is found using the Pythagorean theorem a² + b²  = c²,  or in this case c²-a² =b².

 

(12.5)² - (7.5701)² = h²

 

98.943 = h²       h = 9.947

 

V = (5/6)abh

 

V = (5/6) (7.5701)(11)(9.947)

 

V = 690.248 cubic whatevers