Area of Shapes Explained — A Complete Guide

Area answers a very practical question: how much surface does a shape cover? It tells you how much paint a wall needs, how much carpet a room takes, how much grass seed a garden needs, and how much land a plot contains. The idea is always the same — count how many unit squares fit inside the shape — but the shortcut formula changes from shape to shape, because each shape is built differently.

One rule runs through every formula in this guide. Area is always measured in square units: square centimetres, square metres, square kilometres, and so on. That is because area always comes from multiplying a length by a length. If you multiply metres by metres, the answer is square metres, and if you forget the "square" part, the number alone does not really say what you measured.

Rectangles and squares: length times width

A rectangle is the easiest shape to start with, because it is made of neat rows of unit squares. If a rectangle is 6 metres long and 4 metres wide, you can picture 6 squares along the top and 4 rows down, which is 6 times 4, or 24 squares. The formula is simply area equals length multiplied by width.

A square is just a special rectangle where every side is the same length, so its formula is even shorter: area equals side multiplied by side, often written as side squared. A square with sides of 5 metres has an area of 5 times 5, which is 25 square metres. Notice that the same number, 5, plays both roles here, which is why squaring a number and finding the area of a square share the same word.

Rectangles appear everywhere in everyday life — rooms, screens, sheets of paper, fields — so this formula is the one you will use most often. If a shape looks like a rectangle but has a piece cut out of it, split it into two rectangles, find each area separately, and add them together. That splitting trick works for most irregular shapes made of straight sides.

Circles: pi times radius squared

A circle is different because it has no straight sides and no corners to count squares along. Its area depends on one measurement only: the radius, which is the distance from the centre to the edge. The formula is area equals pi multiplied by radius squared, written as pi times r squared.

Pi is a constant, roughly 3.14159, and for most everyday work 3.14 is close enough. The radius squared part means you multiply the radius by itself, not by two. A circle with a radius of 3 metres has an area of pi times 9, which is about 28.27 square metres. Doubling the radius does not double the area: a radius of 6 metres gives pi times 36, about 113.10 square metres, which is four times as much, because the radius is squared.

The most common circle mistake is using the diameter instead of the radius. The diameter is the distance right across the circle, through the centre, and it is twice the radius. So if you are given a diameter of 10 metres, halve it first to get a radius of 5 metres, and then the area is pi times 25, about 78.54 square metres. Using 10 as if it were the radius would give about 314.16 square metres — four times too big.

Triangles: half of the matching rectangle

Every triangle is exactly half of a rectangle with the same base and height, and that fact gives you its formula: area equals one half multiplied by base multiplied by height. The base can be any side you choose, but the height must then be measured at right angles to that base — straight up from the base to the opposite corner, not along a sloping side.

A triangle with a base of 8 metres and a height of 5 metres has an area of one half times 8 times 5, which is 20 square metres. The matching rectangle would be 8 times 5, or 40 square metres, and the triangle fills exactly half of it. That picture is worth remembering, because forgetting the half is the single most common triangle error.

It does not matter which side you call the base, as long as the height matches it. A tall, thin triangle and a short, wide triangle can have exactly the same area if base times height is the same, which surprises many people. Area cares about the product, not about the shape looking big in one direction.

Parallelograms and trapeziums: two more useful formulas

A parallelogram looks like a rectangle pushed sideways, and its formula is reassuringly similar: area equals base multiplied by height, where the height is again the vertical distance between the two parallel sides, not the length of the sloping side. A parallelogram with a base of 7 metres and a height of 4 metres has an area of 28 square metres, exactly like a rectangle of the same base and height, because the pushed-over piece fills the gap on the other side.

A trapezium has one pair of parallel sides, usually called a and b, and its area is the average of those two sides multiplied by the height between them: area equals one half times a plus b, all multiplied by height. For example, a trapezium with parallel sides of 6 metres and 10 metres and a height of 4 metres has an area of one half times 16 times 4, which is 32 square metres. If the two parallel sides were equal, this formula would collapse back into the rectangle formula, which is a good check that it makes sense.

Worked example 1: carpet for a rectangular room

A room measures 4.5 metres by 3.2 metres. Its area is 4.5 times 3.2, which is 14.4 square metres. That is the amount of carpet needed before allowing for waste and cutting, so in practice you would buy a little more than 14.4 square metres. Notice that both measurements were already in metres. If one had been in centimetres, converting it first would have been essential, as the next example shows.

Worked example 2: a circular garden bed, with a unit trap

A circular flower bed has a diameter of 4 metres, so its radius is 2 metres and its area is pi times 4, about 12.57 square metres. Now suppose a rectangular bed measures 2 metres by 50 centimetres. The trap is to multiply 2 by 50 and call the answer 100 square metres. Converted first, 50 centimetres is 0.5 metres, so the real area is 2 times 0.5, which is 1 square metre. Mixing units before multiplying is the fastest way to get an area wildly wrong, and converting everything to the same unit first is the one habit that prevents it.

Worked example 3: a triangular sail

A triangular sail has a base of 6 metres and a height of 4 metres. Its area is one half times 6 times 4, which is 12 square metres. If you forgot the half, you would report 24 square metres and order twice as much sailcloth as needed. A quick sanity check helps here: the triangle must be smaller than the 6 by 4 rectangle around it, so any answer of 24 or more should immediately look suspicious.

Common mistakes to avoid

Using the diameter as the radius for a circle quadruples the answer, because the radius is squared in the formula. Halve the diameter first, every time. Using a sloping side as the height of a triangle or parallelogram gives an answer that is too large, because the true height is the perpendicular distance. Forgetting the half in the triangle formula doubles the answer. Mixing units, such as metres and centimetres, without converting first, produces a number that looks precise and is completely wrong. And finally, giving an area in plain metres instead of square metres hides what was actually measured — the square unit is part of the answer, not decoration.

Frequently asked questions

What is the difference between area and perimeter?

Area measures the surface covered inside a shape, in square units. Perimeter measures the distance around the edge of a shape, in ordinary length units. A long, thin rectangle and a square can have the same perimeter but very different areas, so the two numbers answer different questions.

Why is area always in square units?

Because every area formula multiplies a length by a length, whether directly, as in length times width, or indirectly, as in pi times radius squared. Length times length gives square units by definition, which is why you write square metres rather than metres.

How do I find the area of an irregular shape?

Split it into rectangles, triangles and other shapes you know, find each area separately, and add them together. For a shape with a curved edge, that method gives an estimate, and the smaller the pieces you split it into, the closer the estimate becomes.

Do I need the exact value of pi?

For school and everyday work, 3.14 is usually enough, and the difference from using more digits is small. For a circle with a radius of 3 metres, pi at 3.14 gives 28.26 square metres against 28.27 with more digits — a difference of one hundredth of a square metre.

If you want to check a result, try the matching tool on this site: the Area Calculator handles rectangles, circles and triangles from the same measurements used above.