Lesson 11: Area and Perimeter

You are buying fencing for a garden and carpet for a bedroom. The fencing is sold by the metre and wraps around the edge of the garden; the carpet is sold by the square metre and covers the floor. Two different measurements, two different jobs: perimeter measures the boundary of a shape, and area measures the surface inside it. Confusing them is expensive — this lesson makes sure you never do, with rectangles, squares, triangles and parallelograms all worked through step by step.

Perimeter: the distance around

The perimeter of a shape is the total length of its boundary. For a rectangle with length l and width w, opposite sides are equal, so the perimeter is P = 2l + 2w. For a square with side s, all four sides match: P = 4s. Perimeter is a length, so it is measured in plain units: metres, feet, centimetres.

Worked example 1. A rectangle is 9 cm long and 6 cm wide. P = 2 × 9 + 2 × 6 = 18 + 12 = 30 cm. Its area (next section) is 9 × 6 = 54 cm² — keep the two answers side by side and notice how different they feel: 30 cm of border, 54 square centimetres of surface.

Worked example 2. A square tile has side 7 cm. P = 4 × 7 = 28 cm; A = 7 × 7 = 49 cm².

Area: covering the inside

The area of a rectangle is A = l × w — the number of unit squares needed to tile it. A square's area is A = s². Because you are multiplying two lengths, area is measured in square units: cm², m², ft². A triangle is exactly half of a rectangle with the same base and height, so its area is A = (b × h) ÷ 2. A parallelogram slants, but if you snip a triangle off one end and reattach it, it becomes a rectangle, so its area is A = b × h — using the vertical height, not the slanted side.

Worked example 3. A triangular flag has base 10 cm and height 6 cm. A = (10 × 6) ÷ 2 = 60 ÷ 2 = 30 cm². Forgetting to halve is the most common triangle error in existence — the ÷ 2 is the whole point.

Worked example 4. A parallelogram has base 8 m and vertical height 5 m. A = 8 × 5 = 40 m², even if its slanted side measures 6 m. The slant is a decoy; only base and perpendicular height count.

In the real world: fencing and carpet

A rectangular garden measures 12 m by 5 m. Fencing it costs $4 per metre: the perimeter is 2 × 12 + 2 × 5 = 34 m, so fencing costs 34 × 4 = $136. A bedroom 4.5 m by 3 m needs carpet covering 4.5 × 3 = 13.5 m². Paint tins state their coverage in square metres (area), picture frames are sold by the size of their border (perimeter), and a running track is measured around its edge (perimeter) while the field inside is mowed by area. Ask "am I going around it or covering it?" and you will always pick the right formula.

Common mistakes

• Using the same formula for both. Perimeter adds side lengths; area multiplies them. 9 × 6 rectangle: perimeter 30 cm, area 54 cm² — different numbers, different units.
• Forgetting the ÷ 2 for triangles. (10 × 6) = 60 is the rectangle's area. The triangle is half: 30 cm².
• Using the slant height of a parallelogram. Area needs the perpendicular (vertical) height. The slanted side is used for perimeter, not area.
• Wrong units. Perimeter in cm² or area in cm should ring alarm bells: lengths are plain units, areas are square units.

Exam Experience

1. What is the area of a triangle with base 12 cm and height 5 cm? A) 17 cm² B) 30 cm² C) 60 cm² D) 120 cm²

2. What is the perimeter of a rectangle 8 m long and 3 m wide? A) 11 m B) 22 m C) 24 m D) 48 m

3. A square has an area of 81 cm². What is the length of one side? A) 9 cm B) 20.25 cm C) 40.5 cm D) 81 cm

4. (Constructed response) A rectangular yard measures 15 ft by 10 ft. In one corner sits a square sandbox that is 3 ft by 3 ft; the rest of the yard is grass. (a) Find the perimeter of the whole yard (for fencing). (b) Find the area of the whole yard. (c) Find the area of grass that needs mowing, explaining your method.

Answer key and explanations

• 1. B — (12 × 5) ÷ 2 = 60 ÷ 2 = 30 cm².
• 2. B — 2 × 8 + 2 × 3 = 16 + 6 = 22 m.
• 3. A — Side = √81 = 9 cm.
• 4. (a) 2 × 15 + 2 × 10 = 50 ft. (b) 15 × 10 = 150 ft². (c) Sandbox area = 3 × 3 = 9 ft², so grass = 150 − 9 = 141 ft² — subtract the part that is not grass from the whole.

Scoring guide for question 4 (3 points)

• 1 point for the perimeter, 50 ft, from adding all four sides (or 2l + 2w).
• 1 point for the yard's area, 150 ft², with square units.
• 1 point for the grass area, 141 ft², found by subtracting the sandbox's 9 ft² — with the method stated, not just the answer.

Going further: composite shapes and tiling

Real floors are rarely one clean rectangle. The strategy for composite shapes: either split the shape into rectangles you can handle, or complete it into one big rectangle and subtract the missing piece.

Worked example 5. An L-shaped room is a 10 m × 8 m rectangle with a 4 m × 3 m corner missing. Complete-and-subtract: 10 × 8 − 4 × 3 = 80 − 12 = 68 m². (Splitting it into a 10 × 5 strip plus a 6 × 3 strip gives 50 + 18 = 68 as well — two methods, one answer, exactly the cross-check professionals rely on.)

Worked example 6. A parallelogram has base 9 cm, slanted side 5 cm and vertical height 7 cm. Its perimeter uses the sides: 2 × 9 + 2 × 5 = 28 cm. Its area uses base × vertical height: 9 × 7 = 63 cm². One shape, both measurements, each using the numbers meant for it — read the question twice to see which job is being asked.

Worked example 7 (tiling). A kitchen floor is 6 m × 4.5 m, to be tiled with square tiles of side 0.5 m. Floor area: 27 m². Tiles along the length: 6 ÷ 0.5 = 12; along the width: 4.5 ÷ 0.5 = 9. Total: 12 × 9 = 108 tiles (plus a few spares for cuts and breakages — tilers order about 10% extra, which is lesson 5's percent skills coming back to work).

Worked example 8 (working backwards). A triangle has area 24 cm² and base 8 cm. Since A = (b × h) ÷ 2, the height is 2 × 24 ÷ 8 = 6 cm. Undoing a formula is a theme of this course: divide where the formula multiplies, and here also multiply where it divides.

Quick practice

• 1. Area of a rectangle 11 m × 4 m.
• 2. Perimeter of a square with side 13 cm.
• 3. Area of a triangle with base 14 cm and height 9 cm.
• 4. Area of a parallelogram with base 12 m and height 6 m.

Answers. 1. 44 m². 2. 4 × 13 = 52 cm. 3. (14 × 9) ÷ 2 = 63 cm². 4. 12 × 6 = 72 m² (no halving for parallelograms — the halve belongs to triangles).

Investigation: painting the bedroom wall

One wall of a bedroom is 5 m wide and 2.6 m high, with a window 1.5 m by 1 m in it. The paintable area is the wall minus the window: 5 × 2.6 − 1.5 × 1 = 13 − 1.5 = 11.5 m². The paint tin promises coverage of 8 m² per litre, so you need 11.5 ÷ 8 = 1.4375 litres — and since paint is sold in whole litres, you buy 2 litres, rounding up because running out mid-wall is worse than having leftovers. Skirting board runs along the bottom of the wall: that is a length, 5 m — but if you trim around the whole room's walls, you are back to perimeter thinking. Notice the workflow, because it is the workflow of every tradesperson: draw the shape, mark what is not painted, compute the whole area, subtract the exclusions, convert to purchasable units, round in the direction the situation demands. Area problems in exams are this exact story with the numbers changed. If you can tell the story, you can do the sum.

Challenge corner

A square has a perimeter of 36 cm. Work backwards: each side is 36 ÷ 4 = 9 cm, so its area is 9 × 9 = 81 cm². Now compare with the square of side 4 from the Try it: perimeter 16 and area 16 — numerically equal, a famous coincidence that only happens at side 4 (perimeter 4s equals area s² only when s = 4). For every other square the two measures part ways, and their units (cm versus cm²) were never the same thing anyway. Finish with a fractional answer, which geometry produces without apology: a triangle with base 7 cm and height 3 cm has area (7 × 3) ÷ 2 = 10.5 cm². Decimal and fractional areas are normal in real measurement — the insistence on whole-number answers is a habit from easier worksheets, not from the world.