Area of Rectangles, Triangles, Parallelograms and Trapezia

In the exam you are often asked to work out the area of a rectangle, triangle, parallelogram or trapezium from a labelled diagram. These questions are usually worth 1 or 2 marks.

Harder questions, worth 3 to 5 marks, ask you to work backwards from an area to a missing length, to compare the areas of two shapes, or to work out how much paint, turf or tiles to buy and what it costs.

The formulae sheet in recent exams gives the formula for the area of a trapezium. The formulae for rectangles, triangles and parallelograms are not on it, so learn them.

Area is how much flat surface a shape covers. Imagine covering the shape with squares that are 1 cm long and 1 cm wide. Each one has an area of 1\text{ cm}^2, which is read as one square centimetre.

Perimeter is the distance all the way round the edge, in units such as cm or m. Area is the space inside, in square units such as \text{cm}^2 or \text{m}^2. Read the question carefully to see which one it asks for.

Every length must be in the same unit before you multiply. If one length is in centimetres and another is in metres, change one of them first.

A diagonal cuts a rectangle into two equal triangles. Each triangle has the same base and height as the rectangle, so each triangle has half the rectangle's area.

So the area of a triangle is

A=\frac12\times b\times h

where b is the base and h is the perpendicular height.

The perpendicular height meets the base at a right angle and goes up to the opposite corner. On a diagram it is often a dashed line with a small square where it meets the base. A sloping side is not the height.

A parallelogram has two pairs of parallel sides. Cut a triangle off one end and move it to the other end. The shape becomes a rectangle with the same base and the same perpendicular height.

No area is lost or gained, so the area of a parallelogram is

A=b\times h

where b is the base and h is the perpendicular height. There is no halving for a parallelogram.

A trapezium has one pair of parallel sides. On a diagram they are marked with arrows. Call their lengths a and b. The height h is the perpendicular distance between them.

The area of a trapezium is

A=\frac12(a+b)h

Add the two parallel sides, multiply by the height, then halve.

Why does it work? Two copies of the trapezium, one turned upside down, fit together to make a parallelogram with base a+b and height h. One trapezium is half of it.

Only the two parallel sides go in the bracket. The sloping sides are not used.

Sometimes you are given the area and asked for a length. Write down the formula, put in the numbers you know, then undo the steps one at a time.

Undo multiplying by dividing. For a triangle or a trapezium, double the area first to undo the halving.

For a trapezium, doubling the area and dividing by the height gives the two parallel sides added together. For example, a trapezium has area 45\text{ cm}^2 and height 5 cm. Then 45\times2\div5=18, so its parallel sides add up to 18 cm.

A missing length is measured in units such as cm or m, not in square units.

• Adding or multiplying all four sides of a rectangle. Multiply the length by the width, just once.
• Using a sloping side as the height. Use the height that meets the base at a right angle.
• Forgetting to halve for a triangle. Base times height is the rectangle, so halve it.
• Putting the wrong sides into the trapezium formula. Only the two parallel sides go in the bracket.
• Leaving out the square units. An area is in units such as \text{cm}^2 or \text{m}^2, but a length is in cm or m.
• Mixing centimetres and metres. Change every length to the same unit before you multiply.