New
New
Year 11
Higher

The area of any triangle

I can derive the formula for the area of any triangle.

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New
New
Year 11
Higher

The area of any triangle

I can derive the formula for the area of any triangle.

Link copied to clipboard

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Lesson details

Key learning points

  1. The formula for the area of a right-angled triangle is 0.5 x base x perpendicular height
  2. With an equilateral or isosceles triangle, you can use Pythagoras' theorem to find the height
  3. If you know one of the base angles, you can use the sine ratio to find the perpendicular height
  4. Doing this leads to the formula 0.5absinC

Keywords

  • Area - The area is the size of the surface and states the number of unit squares needed to completely cover that surface.

  • Sine function - The sine of an angle (sin(θ°)) is the y-coordinate of point P on the triangle formed inside the unit circle.

Common misconception

Using any two side lengths for the sine formula.

It must be two side lengths and the angle between them. So any two side lengths of the triangle can be used so long as you also use the angle between them.


To help you plan your year 11 maths lesson on: The area of any triangle, download all teaching resources for free and adapt to suit your pupils' needs...

You may wish to challenge your pupils to derive the sine formula by themselves before showing them how to derive the formula.
Teacher tip

Licence

This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

Lesson video

6 Questions

Q1.
Match the polygon with the formula for calculating its area.
Correct Answer:Square,Area = l2

Area = l2

Correct Answer:Triangle,Area = bh2

Area = bh2

Correct Answer:Parallelogram,Area = b×h

Area = b×h

Correct Answer:Trapezium,Area = 12(a+b)h

Area = 12(a+b)h

Correct Answer:Circle,Area = π×r2

Area = π×r2

Q2.
Match the areas of these triangles with their dimensions.
Correct Answer:Area = 15 cm2,A triangle with perpendicular height = 5 cm and base = 6 cm

A triangle with perpendicular height = 5 cm and base = 6 cm

Correct Answer:Area = 40 cm2,A triangle with perpendicular height = 10 cm and base = 8 cm

A triangle with perpendicular height = 10 cm and base = 8 cm

Correct Answer:Area = 6 cm2,A triangle with perpendicular height = 3 cm and base = 4 cm

A triangle with perpendicular height = 3 cm and base = 4 cm

Correct Answer:Area = 2 cm2,A triangle with perpendicular height = 2 cm and base = 2 cm

A triangle with perpendicular height = 2 cm and base = 2 cm

Q3.
A regular hexagon is made up of 6 equilateral triangles. The total area of the regular hexagon is 120 cm2. The area of one triangle is cm2.
Correct Answer: 20
Q4.
A right-angled triangle has a height of 8 cm and a base of 5 cm. Work out the length of the hypotenuse to 1 decimal place.
Correct Answer: 9.4 cm, 9.4
Q5.
A right-angled triangle has a height of 4 cm and a hypotenuse of 15 cm. Work out the length of the base to 1 decimal place.
Correct Answer: 14.5 cm, 14.5
Q6.
A rectangle has a width of 5 cm and a diagonal length of 13 cm. The area of the rectangle is cm2.
Correct Answer: 60

6 Questions

Q1.
Match the areas of these triangles with their dimensions.
Correct Answer:Area = 45 cm2,A triangle with perpendicular height = 10 cm and base = 9 cm

A triangle with perpendicular height = 10 cm and base = 9 cm

Correct Answer:Area = 9 cm2,A triangle with perpendicular height = 6 cm and base = 3 cm

A triangle with perpendicular height = 6 cm and base = 3 cm

Correct Answer:Area = 1 cm2,A triangle with perpendicular height = 1 cm and base = 2 cm

A triangle with perpendicular height = 1 cm and base = 2 cm

Correct Answer:Area = 2 cm2,A triangle with perpendicular height = 2 cm and base = 2 cm

A triangle with perpendicular height = 2 cm and base = 2 cm

Q2.
A right-angled triangle has a height of 13 cm and a hypotenuse of 20 cm. The area of the triangle to 1 decimal place is cm2.
Correct Answer: 98.8
Q3.
A triangle has two adjacent lengths of 8 cm and 9 cm with an angle in between of 53°. The area of the triangle to 1 decimal place is cm2.
Correct Answer: 28.8
Q4.
Area = 12absinC, despite a, b and C being explicit in the formula, they just mean two adjacent sides and the angle between them. Select other formulas for the area.
An image in a quiz
Correct answer: Area = 12acsinB
Correct answer: Area = 12bcsinA
Area = 12absinB
Area = 12bcsinC
Area = 12acsinA
Q5.
Match the areas with the dimensions of the triangle.
Correct Answer:23.0 cm2,The adjacent lengths are 7 cm and 9 cm with a 47° angle in between.

The adjacent lengths are 7 cm and 9 cm with a 47° angle in between.

Correct Answer:26.3 cm2,The adjacent lengths are 12 cm and 6 cm with a 47° angle in between.

The adjacent lengths are 12 cm and 6 cm with a 47° angle in between.

Correct Answer:6 cm2,The adjacent lengths are 3 cm and 4 cm with a 90° angle in between.

The adjacent lengths are 3 cm and 4 cm with a 90° angle in between.

Correct Answer:2.4 cm2,The adjacent lengths are 1 cm and 6 cm with a 53° angle in between.

The adjacent lengths are 1 cm and 6 cm with a 53° angle in between.

Correct Answer:93 cm2,The adjacent lengths are 6 cm and 6 cm with a 60° angle in between.

The adjacent lengths are 6 cm and 6 cm with a 60° angle in between.

Q6.
Six equilateral triangles are put together to make a regular hexagon. The length of each triangle is 6 cm. Work out the exact area of the regular hexagon.
Correct answer: 543 cm2
63 cm2
363 cm2
93.5 cm2
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