Area

Area is the amount of flat space a shape covers. You can picture it as the number of unit squares that fit inside the shape with no overlaps and no gaps.

Common units are square centimetres ($\text{cm}^{2}$), square metres ($\text{m}^{2}$) and square kilometres ($\text{km}^{2}$). The little $^{2}$ is essential — area is always in square units. Writing "cm" instead of "cm²" loses a mark.

The rectangle in the diagram is $5$ wide and $3$ tall, so it covers $5 \times 3 = 15$ unit squares. This is the simplest area rule: $\text{area of a rectangle} = \text{length} \times \text{width}.$

Every other area formula in this note comes from rearranging this idea. Cutting and sliding a shape can turn it into a rectangle without losing any area at all.

Two identical triangles fit together to make a parallelogram, and any parallelogram has the same area as a rectangle with the same base and height. So a triangle has half the area of that rectangle: $\text{area of a triangle} = \tfrac{1}{2} \times \text{base} \times \text{perpendicular height}.$

The word "perpendicular" matters. The height $h$ is the straight-line distance from the apex (top corner) to the base, measured at $90\degree$. For a right-angled triangle this is one of the legs; for any other triangle, $h$ is the dashed line shown in the diagram.

If a triangle has base $12$ cm and perpendicular height $5$ cm: $\text{area} = \tfrac{1}{2} \times 12 \times 5 = 30 \;\text{cm}^{2}.$

The formula always works, even when the triangle is obtuse and the perpendicular height falls outside the triangle.

A parallelogram has two pairs of parallel sides. If you cut a triangle off one end and slide it to the other end, you turn the parallelogram into a rectangle with the same base and height. So: $\text{area of a parallelogram} = \text{base} \times \text{perpendicular height}.$

A trapezium has exactly one pair of parallel sides. Stand two identical trapezia next to each other (one flipped upside down) and you get a parallelogram with base $(a + b)$ and height $h$. The trapezium has half that area: $\text{area of a trapezium} = \tfrac{1}{2}(a + b) \times h.$

Here $a$ and $b$ are the two parallel sides and $h$ is the perpendicular distance between them.

For example, a parallelogram with base $10$ cm and perpendicular height $4$ cm has area $10 \times 4 = 40 \;\text{cm}^{2}$. A trapezium with parallel sides $6$ cm and $10$ cm and height $4$ cm has area $\tfrac{1}{2}(6 + 10) \times 4 = \tfrac{1}{2} \times 16 \times 4 = 32 \;\text{cm}^{2}$.

For a circle of radius $r$: $\text{area} = \pi r^{2}.$

The radius is the distance from the centre to the edge. The diameter is the distance straight across through the centre, so $d = 2r$ and $r = \dfrac{d}{2}$.

A common slip is to put the diameter into the formula instead of the radius. If a circle has diameter $14$ cm, then $r = 7$ cm and $\text{area} = \pi \times 7^{2} = 49\pi \approx 153.9 \;\text{cm}^{2}.$

You can leave the answer in terms of $\pi$ (as $49\pi$) or use a calculator value of $\pi \approx 3.142$ for a decimal answer. Always read the question to see which form is wanted.

For a semicircle (half a circle), the area is $\tfrac{1}{2} \pi r^{2}$; for a quarter circle, it is $\tfrac{1}{4} \pi r^{2}$.

A compound shape is built from simpler shapes joined together. Two ideas handle almost every example:

1. Split the shape into rectangles, triangles, trapezia or parts of circles, find the area of each, and add.
2. Subtract a missing piece from a larger shape, for example a circle cut out of a square.

A reliable method:

1. Draw any extra dashed lines that split the shape into pieces you know how to handle.
2. Label every side you need; work out any missing lengths from the totals.
3. Find each piece's area with the right formula.
4. Add the parts (for split) or subtract the cut-out (for subtract).
5. Give the final answer in square units.

For instance, an L-shape made from a $10 \times 7$ rectangle with a $4 \times 3$ corner cut off has area $10 \times 7 - 4 \times 3 = 70 - 12 = 58 \;\text{cm}^{2}$.

A shape might also include part of a circle — a semicircle on top of a rectangle, say — in which case you handle that piece with $\tfrac{1}{2} \pi r^{2}$ and add it to the rectangle.

Strong area skills now open up several big topics later.

• Volume and capacity — many volume formulae start with an area: a prism's volume is its cross-sectional area times its length.
• Surface area — adding up the area of every face of a 3D shape.
• Scale drawings and maps — area scales by the square of the linear scale factor.
• Real-world problems — painting walls, laying carpet, tiling floors, planting a garden, sizing solar panels.
• Statistics — histograms use area, not just height, to represent frequency.

If a later topic feels tough, the chances are it leans on the area formulae here. Keep these formulae and the perpendicular-height rule close at hand.