Perimeter and Area of Plane Shapes

Measuring the boundary and the space inside

Perimeter is the total distance around a shape’s outside boundary. Area is the amount of space inside that boundary. To picture the difference, imagine a rectangular garden: fencing follows its perimeter, while turf covers its area.

Perimeter is measured in length units, such as cm or m. Area is measured in square units, such as $\text{cm}^2$ or $\text{m}^2$. One square centimetre is the area of a square measuring 1 cm by 1 cm.

Lengths must be in the same unit before you combine them. For example, a rectangle measuring 2 m by 50 cm should be treated as 2 m by 0.5 m, or as 200 cm by 50 cm.

Rectangles: adding sides and counting squares

A rectangle has equal opposite sides. If its length is $l$ and its width is $w$, its perimeter is

$$P=l+w+l+w=2(l+w).$$

Its area is

$$A=lw.$$

The area formula comes from counting rows of unit squares: the number of squares in each row is multiplied by the number of rows.

For example, a rectangle measuring 8 cm by 3 cm has perimeter $2(8+3)=22\text{ cm}$ and area $8\times3=24\text{ cm}^2$. These answers measure different things, so they have different units.

A square is a rectangle with all four sides equal. For side length $s$, its perimeter is $4s$ and its area is $s^2$.

Circles: radius, diameter and π

Every point on a circle is the same distance from its centre. That distance is the radius, $r$. The diameter, $d$, crosses the circle through its centre, so .

The perimeter of a circle is called its circumference, $C$. For every circle, circumference divided by diameter gives the same number, $\pi$. Therefore,

$$C=\pi d=2\pi r.$$

The area of a circle is

$$A=\pi r^2.$$

Notice how the radius is used differently: circumference involves $r$, whereas area involves $r^2$. Doubling the radius doubles the circumference but makes the area four times as large.

Suppose a circle has diameter 10 cm. Its radius is $10\div2=5\text{ cm}$. Its circumference is $10\pi\text{ cm}$, and its area is $\pi\times5^2=25\pi\text{ cm}^2$.

These are exact answers. If decimal answers are required, they are approximately 31.4 cm and $78.5\text{ cm}^2$, respectively, to one decimal place.

Base and perpendicular height

For triangles, parallelograms and trapezia, the height must be measured at right angles to the base. A sloping side is not generally the height. The chosen base and its height must belong together.

A parallelogram has two pairs of parallel opposite sides. Cutting a triangular piece from one end and moving it to the other makes a rectangle with the same base and perpendicular height. No area is gained or lost, so

$$A=bh.$$

A triangle occupies half the area of a parallelogram with the same base and perpendicular height:

$$A=\frac12bh.$$

For example, with base 9 cm and perpendicular height 4 cm, a parallelogram has area $9\times4=36\text{ cm}^2$, while a triangle has area $\frac12\times9\times4=18\text{ cm}^2$.

In a right-angled triangle, the two sides meeting at the right angle can be used as the base and perpendicular height. The side opposite the right angle is not the height for either of those bases.

Trapezia: using both parallel sides

A trapezium has one pair of parallel sides. Call their lengths $a$ and $b$, and let $h$ be the perpendicular distance between them. Its area is

$$A=\frac12(a+b)h.$$

You can understand this as the average length of the parallel sides multiplied by the height. Alternatively, two identical trapezia can fit together to make a parallelogram with base $a+b$ and height $h$; one trapezium has half its area.

For parallel sides of 30 cm and 15 cm, separated by a perpendicular height of 20 cm, the area is $\frac12(30+15)\times20=450\text{ cm}^2$.

Combining shapes and finding missing lengths

A more complicated shape can often be split into rectangles, triangles, parallelograms or trapezia. Calculate each part’s area and add the results, ensuring that the parts neither overlap nor leave gaps. If a piece has been removed from a larger shape, subtract its area instead.

For the pentagon shown, a horizontal dividing line separates a rectangle from a right-angled triangle. The rectangle measures 12 cm by 4 cm. The triangle’s base is the remaining horizontal distance, $12-5=7\text{ cm}$, and its height is the extra vertical distance, $9-4=5\text{ cm}$.

The total area is therefore

$$A=12\times4+\frac12\times7\times5=48+17.5=65.5\text{ cm}^2.$$

The dividing line helps calculate area, but it is not part of the outside boundary. When calculating a perimeter, trace only the outside edges and add their lengths.

Rectangles

For length $l$ and width $w$:

• Perimeter: $P=2(l+w)$.