Idiot\'s Guides Basic Math and Pre-Algebra

Chapter 16: Surface Area and Volume 221

It sounds like a lot of work, but here’s how it breaks down. You need to know the perimeter (p)

of the regular polygon at the base, the apothem (a) of that polygon, and the slant height (l) of the

pyramid. Then you can find the surface area.

SA = (^12 × a × p) + (^12 × p × l)

=^12 p(a + l)

To find the surface area of a hexagonal pyramid with a perimeter of 18 inches, an apothem of 2

inches, and a slant height of 8 inches, plug those numbers into the formula.

SA =^12 p(a + l)

=^12 (18)(2 + 8)

= 90

The hexagonal pyramid has a surface area of 90 square inches.

CHECK POINT

Find the surface area of each pyramid. SA =^12 p(a + l)

11. A square pyramid 4 inches on each side with a slant height of 5 inches.


12. A triangular pyramid with a slant of 10 cm, whose base is an equilateral triangle
    12 cm on a side, with an area of 62.4 square centimeters.


13. A pyramid with a slant height of 18 cm and regular pentagon as a base. The reg-
    ular pentagon has a perimeter of 50 cm and an area of 172 square centimeters.


14. A hexagonal pyramid with a slant height of 10 inches. The regular hexagon that
    forms the base has a perimeter of 60 inches and an area of 260 square inches.


15. A square pyramid with a slant height of 13 inches and a side of 10 inches.

Volume

The volume of a prism is V = Bh. The volume of a pyramid with the same base and the same

height must be smaller, because those lateral faces went from rectangles to triangles and tipped

inward. The volume of a pyramid is one-third of the area of the base times the height. If B is the

area of the base and h is the height, the volume is VBh^1

3

.

Let’s return to the hexagonal pyramid you looked at earlier, with p = 18 inches, a = 2 inches, and

l = 8 inches. To find its volume, we need to know the area of the base and its height. (Remember,

the height is different from the slant height.)