Area, Volume & Surface Area

1 · Area formulas

Rectangle: length × width. Parallelogram: base × height. Triangle: ½ × base × height. Trapezoid: ½ × (top + bottom) × height. Circle: πr².

The height is always measured straight up, at a right angle to the base, never along a slanted side.

2 · Composite shapes

Split the shape into pieces you know, or take a big shape and subtract. A 10 by 6 rectangle with a half circle on the 6-inch end: 60 + ½ × π × 3² = 60 + 4.5π.

3 · Shaded regions

Shaded area = whole − the part that is cut out. A 10 by 10 square with a circle that just fits inside: 100 − 25π (the circle's radius is 5, not 10).

4 · Prisms and cylinders

Volume = area of the base × height. A box 5 × 4 × 3 holds 60 cubic units. A cylinder with radius 3 and height 10 holds π × 9 × 10 = 90π.

5 · Cones, pyramids, and spheres

A cone or pyramid holds one third as much as the cylinder or prism with the same base and height: V = ⅓ × base × height. A sphere: V = (4/3)πr³.

Radius 3 and height 8: the cone holds ⅓ × 9π × 8 = 24π.

6 · Surface area of a box

A box has three pairs of matching faces. For 6 × 4 × 2: 2 × (24 + 12 + 8) = 88 square units. A cube with edge s has 6s².

7 · Surface area of a cylinder

Two circles plus the side. The side unrolls into a rectangle as tall as the can and as wide as its circumference. Radius 4, height 5: 2 × 16π + 2π × 4 × 5 = 32π + 40π = 72π.

8 · Scaling

If every length is multiplied by k, areas are multiplied by k² and volumes by k³. Doubling the edges of a cube makes its volume 8 times as big.

For a cylinder, the radius is squared: doubling the radius and halving the height multiplies the volume by 4 × ½ = 2.

9 · Units of area and volume

Square units square the conversion: 1 square yard = 3 × 3 = 9 square feet, and 1 square meter = 100 × 100 = 10,000 square cm.

Cubic units cube it: 1 cubic foot = 12 × 12 × 12 = 1,728 cubic inches. A liter is 1,000 cubic centimeters.

10 · Working backward

If a box holds 72 cubic units and its base is 3 × 4, its height is 72 ÷ 12 = 6. If a cube holds 125 cubic units, its edge is 5, since 5 × 5 × 5 = 125.