7.2 Area & Perimeter

7.2 Area & Perimeter

Quick Answer: Perimeter is the total distance around a shape — add the sides. Area is the space inside a shape — measured in square units. On the NEX you need rectangles, squares, triangles, and circles, plus composite figures built from rectangles.

Perimeter — Linear Units

Perimeter is the sum of the side lengths of a closed shape. It is a length, so its units are linear: inches, feet, meters, centimeters.

Worked example 1. A rectangular clinic room is 14 ft by 10 ft. What is the perimeter?

P = 2 × (14 + 10) = 2 × 24 = 48 ft.

Worked example 2. A triangular dressing covers a wound with sides 5 cm, 6 cm, and 7 cm. Find the perimeter.

P = 5 + 6 + 7 = 18 cm.

When side lengths have mixed units (for example, 2 ft and 18 in), convert to a single unit before adding.

Area — Square Units

Area measures the surface inside a shape. Because it counts squares, the units are always square: in², ft², cm², m². A common NEX trap is reporting area with a linear unit.

Worked example 3. A hospital blanket is 60 in long and 36 in wide. What is its area?

A = 60 × 36 = 2,160 in².

Worked example 4. A triangular bandage has base 12 cm and height 5 cm. Find the area.

A = ½ × 12 × 5 = ½ × 60 = 30 cm². The height here is the perpendicular height, not a slanted side.

Worked example 5. A circular wound has diameter 6 cm. Find the area.

Radius r = 6 ÷ 2 = 3 cm. A = π × r² = 3.14 × 3² = 3.14 × 9 = 28.26 cm².

Circumference — The Circle's Perimeter

Circumference is the perimeter of a circle:

• C = π × d, or
• C = 2 × π × r

Worked example 6. A circular handrail has radius 7 m. Find the circumference (use π ≈ 3.14).

C = 2 × 3.14 × 7 = 43.96 m.

Composite Figures

A composite figure is made of two or more basic shapes. Break it into rectangles (and sometimes triangles), find each area, then add (or subtract for cut-outs).

Worked example 7. An L-shaped exam room can be split into two rectangles: 8 ft × 6 ft and 4 ft × 4 ft. Find the total floor area.

Rectangle 1: 8 × 6 = 48 ft². Rectangle 2: 4 × 4 = 16 ft². Total = 48 + 16 = 64 ft².

Worked example 8. A rectangular floor 10 m by 7 m has a circular utility cut-out of radius 1 m. What is the usable floor area?

Floor area = 10 × 7 = 70 m². Cut-out = π × 1² = 3.14 m². Usable = 70 − 3.14 = 66.86 m².

Mixed Units — Convert First

Always check unit consistency before substituting into a formula.

Worked example 9. A rectangle is 1.2 m by 80 cm. Find the area in cm².

Convert 1.2 m = 120 cm. A = 120 × 80 = 9,600 cm².

If the question wants the answer in m² instead: 1.2 m × 0.8 m = 0.96 m². Same rectangle, different unit — both are correct.

Nursing-Relevant Examples

• Wound/dressing area — size a gauze pad to cover a rectangular wound 3 cm × 4 cm: area = 12 cm².
• Floor area for equipment — check whether a 2 ft × 3 ft crash cart fits in a 6 ft × 8 ft alcove: alcove area = 48 ft², cart footprint = 6 ft², so it fits.
• Circular basin — estimate the surface area of a basin with radius 5 in: A = 3.14 × 25 = 78.5 in².

Common Traps

• Using diameter as radius. Circle area uses r, not d. If you are given the diameter, halve it first.
• Forgetting the ½ in triangle area. A = ½ × b × h, not b × h.
• Area vs. perimeter units. Perimeter uses linear units (cm, m, in); area uses square units (cm², m², in²). A number followed by a linear unit when the question asked for area is a red flag.
• Using slant height for triangle area. The height in ½ × b × h is perpendicular to the base, not a side length unless the triangle is right-angled.
• Mixing units without converting. A rectangle 1.5 m × 60 cm is not 1.5 × 60 = 90. Convert one side first.