Area, Volume & Units

1 · Base and height

For parallelograms, triangles, and trapezoids, the height is the straight-up distance, at a right angle to the base. It is often drawn as a dashed line. The slanted side is not the height.

Parallelogram: base × height. Triangle: ½ × base × height. Trapezoid: ½ × (top + bottom) × height. A trapezoid with parallel sides 6 and 10 and height 4 has area ½ × 16 × 4 = 32.

2 · Composite figures

Break an odd shape into rectangles, triangles, and parts of circles, then add. If a piece is cut out, subtract it.

A 10 by 8 rectangle with a triangle (base 4, height 3) cut from one edge has area 80 − 6 = 74.

3 · Volume of prisms

The volume of any prism is the area of its base times its height. For a box, that is length × width × height: 6 × 4 × 5 = 120 cubic units.

A fish tank 50 cm long and 30 cm wide filled to 20 cm holds 50 × 30 × 20 = 30,000 cubic centimeters of water, which is 30 liters.

4 · Volume of cylinders

A cylinder is a prism with a circle for a base: V = πr²h. A can with radius 3 and height 10 holds π × 9 × 10 = 90π cubic units.

Using 2πr (the circumference) instead of πr² is the usual slip.

5 · Surface area

Surface area is the total area of the outside. A box has three pairs of matching faces: SA = 2(lw + lh + wh). A 5 by 3 by 2 box: 2(15 + 10 + 6) = 62 square units.

A box with no lid has one face missing, so subtract it. A cube has 6 equal square faces: 6s².

6 · When the edges change

If only one dimension is doubled, the volume doubles. If every edge is doubled, the volume is multiplied by 2 × 2 × 2 = 8, and the surface area by 4.

7 · Square and cubic units

1 yard = 3 feet, but 1 square yard = 3 × 3 = 9 square feet, and 1 cubic yard = 3 × 3 × 3 = 27 cubic feet.

So 4 square yards of carpet is 36 square feet, not 12.

8 · Converting rates

Convert one unit at a time. 30 miles per hour into feet per second: 30 × 5,280 = 158,400 feet per hour; 158,400 ÷ 3,600 = 44 feet per second.

Write the units as you go, and they will tell you whether to multiply or divide.

9 · The metric ladder

kilo- = 1,000, centi- = 1/100, milli- = 1/1,000. 2.5 kilometers = 2,500 meters = 250,000 centimeters.

Going to a smaller unit, the number gets bigger (multiply). Going to a bigger unit, the number gets smaller (divide).

10 · On the ISEE

Check that the answer has the right kind of unit: a length in units, an area in square units, a volume in cubic units. Many wrong choices are the right number for the wrong measure, like the perimeter when the area is asked.