4.2 Perimeter, area, surface area & volume

Perimeter, Area, Surface Area, and Volume

These four measurements answer different questions. Perimeter measures the distance around a flat shape (one dimension, units like ft). Area measures the flat space inside a shape (two dimensions, square units like ft²). Surface area adds up the areas of every face of a solid (still square units). Volume measures the space a solid holds (three dimensions, cubic units like ft³). Matching the question to the correct formula — and the correct kind of unit — is half the battle on the TSIA2.

Formula reference table

Use π ≈ 3.14 unless a problem tells you otherwise.

Perimeter and circumference

Worked example 1: a rectangle is 9 ft long and 4 ft wide. P = 2(9) + 2(4) = 18 + 8 = 26 ft.

Worked example 2: a circle has radius 5 cm. C = 2πr = 2 × 3.14 × 5 = 31.4 cm. If instead you are given the diameter of 10 cm, C = πd = 3.14 × 10 = 31.4 cm — the same result, because the diameter is twice the radius.

Area

Worked example 3 (triangle): base 12 in, height 7 in. A = ½ × 12 × 7 = ½ × 84 = 42 in².

Worked example 4 (circle): radius 6 m. A = πr² = 3.14 × 6² = 3.14 × 36 = 113.04 m². Square the radius before multiplying by π; squaring 6 gives 36, not 12.

Worked example 5 (trapezoid): parallel sides 8 and 14 cm, height 5 cm. A = ½ (8 + 14) × 5 = ½ × 22 × 5 = 11 × 5 = 55 cm².

Composite figures

Break a compound shape into familiar pieces, find each area, then add them — or subtract a cut-out.

Worked example 6: an L-shaped floor is a 10 ft × 6 ft rectangle with a 3 ft × 4 ft rectangle removed from one corner. Area = (10 × 6) − (3 × 4) = 60 − 12 = 48 ft². A "rectangle plus half-circle" window works the same way: add the semicircle's ½πr² to the rectangle's l × w.

Surface area

Surface area is the total "wrapping paper" of a solid. For a box, find each of the three pairs of faces.

Worked example 7: a box measures 5 × 3 × 2 ft. SA = 2(lw + lh + wh) = 2(5×3 + 5×2 + 3×2) = 2(15 + 10 + 6) = 2 × 31 = 62 ft².

For a cylinder, add the two circular ends to the wrapped side. With r = 3 in and h = 10 in: SA = 2πr² + 2πrh = 2(3.14)(9) + 2(3.14)(3)(10) = 56.52 + 188.4 = 244.92 in².

Volume

Worked example 8 (prism): a tank 4 × 3 × 2 m holds V = 4 × 3 × 2 = 24 m³.

Worked example 9 (cylinder): r = 5 cm, h = 8 cm. V = πr²h = 3.14 × 25 × 8 = 3.14 × 200 = 628 cm³.

Worked example 10 (cone): same r and h as the cylinder. A cone is exactly one-third of its matching cylinder, so V = ⅓ × 628 = 209.33 cm³.

Worked example 11 (sphere): r = 3 in. V = 4⁄3 πr³ = 4⁄3 × 3.14 × 27 = 4⁄3 × 84.78 = 113.04 in³.

Matching the question to the formula

The hardest step is often deciding which measurement a word problem wants. "How much fencing surrounds a garden?" asks for perimeter. "How much sod covers a lawn?" asks for area. "How much water fills a pool?" asks for volume. "How much paint coats a storage tank?" asks for surface area. Reading for these cues first tells you the units and the formula before you touch a single number.

Worked example 12: how much carpet covers a 12 ft by 9 ft room? The word "covers" signals area, so A = 12 × 9 = 108 ft². If the same room instead needed baseboard trim around its edges, that would be perimeter: P = 2(12) + 2(9) = 24 + 18 = 42 ft. Same rectangle, two very different questions and two different kinds of units.

Unit sense and reasonableness

Perimeter answers end in plain units (ft), area answers in square units (ft²), and volume answers in cubic units (ft³). If a "volume" comes out in square units, a formula went wrong somewhere. Doubling every dimension of a box multiplies its volume by 2³ = 8, not by 2 — a handy reasonableness check. Finally, keep radius and diameter straight: the radius is half the diameter, and area needs r², so a very common error is forgetting to halve a given diameter before squaring it.