Area of triangles - KS3 Maths

Key points

An image of a square grid. The grid has a length of fourteen squares and a width of eight squares. A rectangle with length four squares and width three squares has been drawn on the grid. A diagonal dashed line joins the top left vertex to the bottom right vertex, splitting the rectangle into two right angled triangles. Each triangle is coloured a different shade of green. — Image caption,
A rectangle split in half creates two triangles.

Any can be shown to be half of a . This can be shown in diagrams and practically using paper and scissors.
The of a triangle is calculated by finding the length of the base, halving it, and then multiplying it by the height. Alternatively, multiply the base length by the perpendicular height and then halve to find the area of a triangle.
The base and the perpendicular height are at right angles to each other. The measurements used must be in the same units. For example, to work in cm a 45 mm measurement would have to be converted to 4۰5 cm. Practising converting metric units will help with this process.

Understanding the formula for the area of a triangle

The area of a triangle is calculated by the : \( \frac{1}{2} \) × base × perpendicular height.

A rectangle can be divided into two triangles.
The length and width of the rectangle are the base and height of a triangle. The base and the height are at right-angles, making the height the perpendicular height.
For a right-angled triangle, the base and the perpendicular height are interchangeable.
The of the triangle means that the base is not necessarily horizontal.
For triangles that are not right-angled, the base is one of the sides and the perpendicular height may be shown either inside or outside the shape, it is not another side.
The base is one side of the triangle. The perpendicular height is the perpendicular measurement that gives the other dimension of the triangle.

The formula can be presented as:

𝑨 = \( \frac{𝒃𝒉}{2} \)or 𝑨 = \( \frac{1}{2} \) 𝒃𝒉

𝑨 is the area of the triangle.
𝒃 is the length of the base of the triangle.
𝒉 is the length of the perpendicular height of the triangle.

Example

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The rectangle has been cut into two congruent triangles by drawing a diagonal.
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The triangles are identical. The area of one triangle is equivalent to half of the area of the rectangle. The area of the rectangle is the length multiplied by the width. The length and width of the rectangle are the base and perpendicular height of the triangle, they are at right-angles to each other. The area of the triangle is half of the base multiplied by the perpendicular height.
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For a right-angled triangle, the base and the perpendicular height are the two sides that are at right angles. The perpendicular height is the perpendicular measurement that gives the other dimension of the triangle. The terms are interchangeable and each of the perpendicular sides can be the base or the height. The orientation of the triangle means that the base is not necessarily horizontal or the lowest side of the shape.
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For triangles that are not right-angled, the base is one of the sides and the perpendicular height may be shown either inside or outside the shape, it is not another side.
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The base is one side of the triangle. The perpendicular height is the perpendicular measurement that gives the other dimension of the triangle.
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A triangle is half of a rectangle. This can be shown in a diagram.
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The area of one triangle is ½ × base × perpendicular height.
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It is not always clear straightaway that a triangle is half of a rectangle. Two identical triangles make a parallelogram, a quadrilateral with opposite pairs of sides that are both equal in length and parallel (always equal distances from each other). The two triangles have the same area, half of the parallelogram.
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The parallelogram can be cut to make a rectangle. The rectangle is made up of the two original triangles. Each triangle is half the area of the rectangle.
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The area of a parallelogram is base × perpendicular height. The area of one triangle is half of the parallelogram, this is ½ × base × perpendicular height. The formula can be written in equivalent forms, 𝑨 = ½ 𝒃𝒉 and 𝑨 = ᵇʰ⁄₂.

Question

Which lettered lengths can be used as the base and perpendicular height for the given triangles?

A series of three images. The first image shows an obtuse angled triangle with sides labelled c, d and e. The perpendicular distance between the side labelled c and the opposite vertex has been drawn, outside the shape, with dashed lines and labelled as a. A further dashed line has been extended in the same direction as side c to this perpendicular label, and labelled b. The second image shows a right angled triangle. The two perpendicular sides are labelled as j and m. The third side is labelled as l. The distance between the right angled vertex and side l has been labelled as k, however this is not perpendicular to side l. The third image shows an acute angled triangle. One side is horizontal and labelled as i. The two other sides are labelled as f and h. The perpendicular distance between side I and the opposite vertex has been labelled as g.

Calculating the area from the base and perpendicular height

The area of a triangle is \( \frac{1}{2} \) × base × perpendicular height. Multiplication is so the calculation can be processed in different orders.

The formula can be presented as:

𝑨 = \( \frac{𝒃𝒉}{2} \)or 𝑨 = \( \frac{1}{2} \) 𝒃𝒉

𝑨 is the area of the triangle.
𝒃 is the length of the base of the triangle.
𝒉 is the length of the perpendicular height of the triangle.

To work out the area of a triangle:

Identify the base and the perpendicular height of the triangle. They must be measured in the same units.
Substitute the base and height into the formula.
Perform the calculation in one of three ways, all of which will give the same answer:
- Multiply the base by the perpendicular height, then divide by two.
- Find half of the base, then multiply by the perpendicular height.
- Find half of the perpendicular height, then multiply by the base.

Examples

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Find the area of the triangle.
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The base and the perpendicular height of the triangle are in the same units. The base of the triangle is 6 cm. The perpendicular height is 8 cm. Substitute the values of the base and the perpendicular height into the formula. 𝒃 = 6 and 𝒉 = 8
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The calculation (6 × 8) ÷ 2 can be worked out in three different ways. Each method gives the same answer of 24. Therefore, the area of the triangle is 24 cm².
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Find the area of this right-angled triangle.
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All three side measurements are given. The longest side of 41 mm is not used because it is not at right angles to either of the other two sides. The sides that are at right angles are 9 mm and 40 mm. One of these is the base and the other is the perpendicular height. If the base is 9 mm the perpendicular height is 40 mm. If the base is 40 mm the perpendicular height is 9 mm. The calculations (9 × 40) ÷ 2 and (40 × 9) ÷ 2 give the same answer. The area of the triangle is therefore 180 mm²
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Work out the area of the triangle.
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All the measurements are in the same units. The base and the perpendicular height are at right angles. The base is 7 cm. The perpendicular height is 4 cm. Substitute the base and perpendicular height values into the formula. (7 × 4) ÷ 2 = 28 ÷ 2 = 14. The area of the triangle is therefore 14 cm²
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Work out the area of the triangle.
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The measurements are not in the same units. The base and the height are at right angles. The base is 2 metres. The height is 400 cm. Convert the units to match, so 400 cm = 4 m
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The base and the height are at right angles. The base is 2 m. The height is 4 m. Substitute the base and perpendicular height values into the formula. (2 × 4) ÷ 2 = 4 and (8 ÷ 2) = 4. The area of the triangle is 4 m²

Question

Work out the area of the triangle.

An image of a right angled triangle. The two perpendicular sides are labelled as lengths nine millimetres and one point two centimetres. The third side is labelled as length one point five centimetres. The triangle is coloured green.

Calculating the base or perpendicular height from the area

To calculate the base of a triangle, choose one of two methods. Both calculations give the same answer.

Method 1: Multiply the area of the triangle by two, then divide by the perpendicular height.

Method 2: Divide the area of the triangle by the perpendicular height, then multiply by two.

To calculate the perpendicular height of a triangle, choose one of two methods. Both calculations give the same answer.

Method 1: Multiply the area of the triangle by two, then divide by the base.

Method 2: Divide the area of the triangle by the base, then multiply by two.

Examples

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The triangle has an area of 72 m². Find the base of the triangle.
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The height of the triangle is perpendicular to the base. The perpendicular height is 8 m. Use one of two calculations to work out the base. Multiply the area of the triangle by two, then divide by the perpendicular height, (72 × 2) ÷ 8. Alternatively, divide the area of the triangle by the perpendicular height, and then multiply by two, (72 ÷ 8) × 2. Both calculations give the same answer. The base of the triangle is 18 m.
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The triangle has a base of 15 mm and an area of 30 mm². From this, find the perpendicular height of the triangle.
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Use one of two calculations to work out the perpendicular height. Multiply the area of the triangle by two, then divide by the base, (30 × 2) ÷ 15 = 60 ÷ 15. Alternatively, divide the area of the triangle by the base, and then multiply by two, (30 ÷ 15) × 2 = 2 x 2. Both calculations give the same answer. The perpendicular height of the triangle is 4 mm.

Question

The triangle has an area of 24 m². Find the perpendicular height of the triangle.

An image of a triangle. The triangle has sides of length four metres, thirteen metres and fifteen metres. The perpendicular distance between the four metre side and the opposite vertex has been drawn, outside the shape, with dashed lines and labelled as question mark. Written below: area equals twenty four metres squared.

Practise finding the area of triangles

Practise finding the area of triangles with this quiz. You may need a pen and paper to help you with your answers.

Quiz

Real-life maths

An image of a triangular shaped flower bed. — Image caption,
A landscape designer would need to calculate the area of a triangle-shaped flowerbed.

A landscape designer may choose to include a triangle-shaped flower bed when creating a new garden layout. To ensure that plants grow well, soil can sometimes require additional nutrients, such as feed or fertiliser.

Nutrients added to soil are used in specific quantities, per unit of area of surface soil.

The landscape designer will need to calculate the area of the triangular flower bed to ensure that the amount of feed or fertiliser used is correct.