If your child gets triangle areas wrong, check whether they use a matching base and perpendicular height, then halve their product. One slip to look for first is halving a shape that should stay whole, like a rectangle. Here is how to find the area of a triangle, and how to tell when the halving step belongs.
The short way
- The area of a triangle is half of base times height.
- The height has to be perpendicular to the base you pick, which means at a right angle to it.
- A rectangle is base times height with no halving, so the halving step is what makes a triangle different.
- Match a chosen base with its perpendicular height. The height may be inside the triangle rather than along a side.
- The same answer comes out whichever matching base and height you choose.
Why this triangle is half its rectangle
Start with an 8 cm by 6 cm rectangle. Its area is 8 times 6, which is 48 cm². Now draw a straight line from one corner to the opposite corner. That diagonal cuts the rectangle into two triangles that are exactly the same size. So each triangle is half of the rectangle. That is where the halving comes from.
So the area of a triangle is half of base times height. You work out base times height, the same as a rectangle, then you halve it. For the triangle inside that 8 by 6 rectangle, that is 48 halved, which is 24 cm².
How to find the area of a triangle
Here is that triangle, drawn inside its rectangle, next to a rectangle that is not halved. The base and the height meet at a right angle.
The triangle: half of 8 × 6 = 24 cm²
The rectangle: 7 × 3 = 21 cm², no halving
Give the units. Area is measured in squares, so the answer is 24 cm², not 24.
This right triangle is half of the rectangle formed by its perpendicular sides. Not to scale.
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Holly, Maths tutor
Holly tutors Maths from KS2 and 11+ entrance preparation through to GCSE and A Level. She has tutored since the age of fifteen, using tailored materials and worked examples.
When do you halve, and when don’t you?
The 7 cm by 3 cm rectangle has area 21 cm², with no halving. First identify the shape: halve base × perpendicular height for a triangle, not a rectangle.
Rotated right triangle. The right angle is at a vertex, between the 6 cm and 8 cm sides. Not to scale.
In the rotated right triangle, the 6 cm and 8 cm sides meet at a right angle. Either can be the base, with the other as its height. Its area is 6 × 8 ÷ 2 = 24 cm². The 10 cm side is not needed for this pair.
Height means the perpendicular distance from the opposite vertex to the line of the chosen base. It is not necessarily vertical on the page, and it need not be a side. Turning a triangle does not change its area.
Acute triangle. The dashed height meets the base at a right angle, inside the base. Not to scale.
In the acute triangle shown, the 10 cm base has a 4 cm perpendicular height. The height splits the base into 1 cm and 9 cm parts. The two right triangles have areas 1 × 4 ÷ 2 = 2 cm² and 9 × 4 ÷ 2 = 18 cm². Together they give 20 cm², the same as 10 × 4 ÷ 2.
The triangles here are ones we drew to show the method, with the lengths written on. They are not to scale, so work from the numbers, not a ruler.
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This clip shows a maths lesson recorded in Pencil Spaces, the online classroom used by Greenhill Academics.
Try this at home this week
Allow about eight minutes. Use the four labelled diagrams below, paper and a pencil. Work from the given lengths; do not measure the pictures.
1
Right triangle.
2
Rectangle.
3
Rotated triangle.
4
Acute triangle.
- For each diagram, ask first: is it a triangle? If it is the rectangle, do base times height and stop. No halving.
- For a triangle, identify the labelled base and matching perpendicular height. The height may be a side or a dashed line inside it.
- Do base times height, then halve it.
- Write the answer with its units (cm²).
The finished answers. Shape 1: 8 × 6 ÷ 2 = 24 cm². Shape 2 (rectangle): 7 × 3 = 21 cm², no halving. Shape 3: 6 × 8 ÷ 2 = 24 cm²; the 10 cm side is not needed. Shape 4: 10 × 4 ÷ 2 = 20 cm².
Watch whether they halve only the triangles and use a complete base with its matching perpendicular height, even when the diagram is rotated.
What this cannot tell you: it only covers triangles where the height is given. It does not cover finding a missing height, or shapes made of several pieces.
Where a tutor takes it next: shapes split into rectangles and triangles, then triangles where the height has to be found.
Our Year 6 maths tutors
Holly tutors Maths from KS2 and 11+ entrance preparation up to GCSE and A Level, including Further Maths. She has tutored since the age of fifteen, with tailored materials and worked examples.
Murray tutors Maths, Chemistry and Physics, from 11+ through GCSE to A Level. He has broad tutoring experience and has run problem-solving sessions for large student groups.
What a tutor does with this
A tutor can check when your child halves and how they match base and height. For families also preparing for the 11+, our separate guide discusses preparation. If timed papers are a concern, read about exam habits to check.
Questions parents ask
My child can do rectangles but gets triangles wrong. Why?
One thing to check is whether they add the halving step for a triangle, and leave a rectangle whole. A triangle is half of base times height, so the extra step is the halve.
Which numbers are the base and the height?
Choose a base and its matching perpendicular height. In a right triangle, the two sides meeting at the right angle can be that pair. In the acute example, use the whole 10 cm base and the dashed 4 cm height, not a sloping side or only part of the base.
Does it matter which side I call the base?
Any side can be chosen as the base if its matching perpendicular height is known. Changing the base does not change the area. In this task, use the base and height supplied in each diagram.
Is this on the Year 6 maths curriculum?
Yes. England’s national curriculum includes calculating triangle areas in Year 6 mathematics.
Do you tutor Year 6 and 11+ maths online?
Yes, we tutor primary and 11+ maths online for families in the UK and abroad. Tell us your child’s year and what they are working towards, and we can talk about the right support.
Want a tutor to work on this with your child?
Tell us your child’s year and school, and we can talk about the right maths support.
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