220 Part 3: The Shape of the World
Surface Area
Like the prism, the pyramid can be broken down into its parts to find the surface area. If the base
has n sides, the surface area is the area of the base plus the area of the n triangles that surround it.
You can calculate each of the pieces separately and add them up.MATH TRAP
You need to have the right slant on measurements in a pyramid. The word “height”
gets used in a lot of different ways and doesn’t always mean the same thing. In a
prism, the height is one of the dimensions of the surrounding rectangles, because
those rectangles stand straight up at right angles to the base. In a prism, the
surrounding triangles are tilted. Their height, called the slant height, is longer than
the height of the pyramid.Usually when you’re asked to find the surface area of a pyramid with a polygon as its base, that
polygon is regular. All its sides are the same length and all its angles are equal. That makes finding
its area a little easier. The area of a regular polygon is^1
2
times the apothem times the perimeter.To find the surface area of a hexagonal pyramid, you have to know the area of the hexagon at the
base and the area of the six surrounding triangles. If the hexagon is a regular hexagon, you can find
the area if you know a side (or the perimeter) and the apothem. To find the area of the triangles,
you need to know the side of the hexagon and the height of the triangle, which is the slant height
of the pyramid. The lateral area is six times the area of one triangle, or^12 times the perimeter times
the slant height.WORLDLY WISDOM
The area of a regular polygon with n sides is the total of the n triangles into which it
is broken when you draw the radii from the center to each vertex. That translates to
n v 21 v apothem v a side. Because n times a side is also the perimeter, the area of a
regular polygon is 21 v apothem v perimeter.