Perimeter and area of rectangles and squares

Rectangles and squares are the most common shapes in everyday life , books, walls, doors, mobile screens, windows, fields. Knowing their perimeter and area is one of the most practically useful skills in mathematics.

Concept

Square

A square has $4$ equal sides and $4$ right angles. If its side is $s$ :

$\text{Perimeter} = 4s \qquad \text{Area} = s^2$

Doubling the side doubles the perimeter, but quadruples the area (since $(2s)^2 = 4s^2$ ).

Rectangle

A rectangle has $4$ right angles, with two pairs of equal opposite sides , length $l$ and breadth $b$ (sometimes called width):

$\text{Perimeter} = 2(l + b) \qquad \text{Area} = l \times b$

If $l = b$ , the rectangle becomes a square. So a square is just a special rectangle where both dimensions are equal.

A surprising fact

Rectangles with the same perimeter can have very different areas. Among all rectangles with a fixed perimeter, the square has the largest area.

Example: perimeter $= 24$ cm.

$1 \times 11$ rectangle: area $= 11$ cm².
$2 \times 10$ : area $= 20$ cm².
$3 \times 9$ : area $= 27$ cm².
$4 \times 8$ : area $= 32$ cm².
$5 \times 7$ : area $= 35$ cm².
$6 \times 6$ (square!): area $= 36$ cm².

The square wins. This is why bees build hexagonal honeycombs (close to circular shapes) rather than long thin tubes , they pack more honey for the same wax.

Similarly, rectangles with the same area can have very different perimeters. A long, thin rectangle has lots of border for little inside; a near-square has the least border for its area.

Combined shapes

Sometimes a real-world shape is two or more rectangles glued together , an L-shape or T-shape. To find the area, split it into separate rectangles, find each area, and add them up. For the perimeter, walk carefully around the boundary, adding each segment exactly once.

Worked examples

Example 1. A rectangle is $25$ m by $14$ m. Find its perimeter and area.

Perimeter $= 2(25 + 14) = 2 \times 39 = 78$ m.
Area $= 25 \times 14 = 350$ m².

Example 2. A square has area $81$ cm². Find its side and its perimeter.

$s^2 = 81 \Rightarrow s = 9$ cm.
Perimeter $= 4 \times 9 = 36$ cm.

Example 3. A rectangular field has perimeter $100$ m and length $30$ m. Find its area.

$2(30 + b) = 100 \Rightarrow 30 + b = 50 \Rightarrow b = 20$ m.
Area $= 30 \times 20 = 600$ m².

Example 4. A wall is $4$ m by $3$ m and has a door $2$ m by $1$ m and a window $1$ m by $1$ m. What is the area to be painted?

Wall area $= 4 \times 3 = 12$ m².
Door $= 2 \times 1 = 2$ m². Window $= 1 \times 1 = 1$ m².
Paintable area $= 12 - 2 - 1 = 9$ m².

Example 5. An L-shaped park is made by joining a $20$ m by $15$ m rectangle and a $10$ m by $8$ m rectangle so that they share a $10$ -m edge (no overlap). Find the total area.

Area $1 = 20 \times 15 = 300$ m².
Area $2 = 10 \times 8 = 80$ m².
Total $= 380$ m².

Example 6. Two rectangles have the same perimeter of $20$ cm. One is $1 \times 9$ and the other is $4 \times 6$ . Which has the larger area?

$1 \times 9 = 9$ cm². $4 \times 6 = 24$ cm². The $4 \times 6$ wins by a lot.

Try it yourself

A rectangle is $18$ cm by $12$ cm. Find its perimeter and area.
A square has perimeter $52$ cm. Find its side and area.
A rectangle has area $48$ cm² and length $8$ cm. Find its breadth and perimeter.
A room is $5$ m by $4$ m. How many $50$ -cm by $50$ -cm tiles will cover the floor?
Find the area of an L-shape made of a $9 \times 4$ rectangle and a $5 \times 3$ rectangle joined at one edge.
Two rectangles have perimeter $24$ cm. One is $3 \times 9$ , the other is $5 \times 7$ . Which has larger area?
A square garden has area $144$ m². Find its perimeter.
Investigate: if you triple the side of a square, what happens to (a) the perimeter, (b) the area?

Activity

Wall paint planner. Pick one wall in your home. Measure its length and breadth in metres. Calculate its area. Now measure any door or window on it and subtract those areas. The number you get is what a painter would actually paint. Double-check by measuring the perimeter too , that is how much border tape would be needed.