Surface Area

Overview

Surface area is the total area of every face of a solid — imagine unfolding it flat and adding up all the pieces. The method never changes: find the area of each face (counting congruent faces more than once), then add. The formulas simply package that counting for prisms, cylinders, pyramids, cones, and spheres.

Vocabulary

Surface Area: The sum of the areas of the faces (surfaces or sides) of a solid figure. Surface area is measured in units squared (units²).
Prism: A polyhedron with two congruent faces, called bases, that lie in parallel planes; the other faces are its lateral faces. Surface area = 2 × (area of a base) + (sum of the areas of the lateral faces).
Slant Height: The height of a triangular lateral face of a pyramid, or the distance from the apex of a cone down its side. Denoted l for a cone.
Cylinder: A solid with two congruent circular bases; its surface area is SA = 2πr² + 2πrh — two circles plus the rectangle formed by unrolling the side.
Sphere: The set of all points in space a fixed distance (the radius) from a center point. Its surface area is SA = 4πr².

Main Ideas

A cube with side 2 has 6 faces of area 4 each, so SA = 24 square inches.
For a prism: two bases plus all the lateral faces. The number of lateral faces equals the number of sides of the base, and they are congruent only when the base is a regular polygon.
Regular pyramid: SA = (area of base) + ½ × (perimeter of base) × (slant height).
Cone: SA = πr² + πrl. The slant height l comes from the Pythagorean Theorem when only the vertical height is given: l = √(r² + h²).
Surface area is in units², never units³ — check the label as well as the number.
Working backwards: a cube's SA divided by 6 gives one face's area, and the square root of that gives the side length.

Lesson Notes

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Practice Problems

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