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Calculating the area of any triangle when the height is not known

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

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New
New

Calculating the area of any triangle when the height is not known

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

Download all resources
Share activities with pupils

These resources will be removed by end of Summer Term 2025.

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Lesson details

Key learning points

  1. The formula for the area of any triangle can be used if the lengths of 2 sides are known
  2. The size of the angle between the two sides must also be known
  3. The formula can be used to find a missing side length or angle if the area is known

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 support pupils, you may wish to have the formula for the area of any triangle displayed separately with a labelled triangle.
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

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Prior knowledge starter quiz

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6 Questions

Q1.
Match the areas of these triangles with their dimensions.
Correct Answer:Area = 15 cm$$^2$$,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 = 22 cm$$^2$$,A triangle with perpendicular height = 11 cm and base = 4 cm

A triangle with perpendicular height = 11 cm and base = 4 cm

Correct Answer:Area = 16 cm$$^2$$,A triangle with perpendicular height = 4 cm and base = 8 cm

A triangle with perpendicular height = 4 cm and base = 8 cm

Correct Answer:Area = 4.5 cm$$^2$$,A triangle with perpendicular height = 3 cm and base = 3 cm

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

Q2.
Which of the following calculate the area of this triangle?
An image in a quiz
Correct answer: $$\frac{1}{2}\times 6.3\times5\times\text{sin}(55.3°)$$
Correct answer: $$\frac{1}{2}\times 6.3\times5.4\times\text{sin}(49.8°)$$
Correct answer: $$\frac{1}{2}\times 5.4\times5\times\text{sin}(74.9°)$$
$$\frac{1}{2}\times 6.3\times5.4\times\text{sin}(55.3°)$$
$$\frac{1}{2}\times 5.4\times5\times\text{sin}(55.3°)$$
Q3.
The area of this triangle, given to 2 significant figures, is cm$$^2$$.
An image in a quiz
Correct Answer: 13
Q4.
A right angled triangle has a height of 12 cm and a hypotenuse of 13 cm. Work out the area.
Correct answer: 30 cm$$^2$$
60 cm$$^2$$
15 cm$$^2$$
120 cm$$^2$$
Q5.
Six equilateral triangles are put together to make a regular hexagon. The length of each triangle is 3 cm. Work out the exact area of the regular hexagon.
Correct answer: $$13.5\sqrt{3}$$ cm$$^2$$
$$27\sqrt{3}$$ cm$$^2$$
$$\sqrt{3}$$ cm$$^2$$
$$27\sqrt{3}$$ cm$$^2$$
Q6.
An isosceles triangle has a base of 4 cm and base angles of 68.2°. Select the correct area given to 2 significant figures.
5.0 cm$$^2$$
Correct answer: 10 cm$$^2$$
20 cm$$^2$$
40 cm$$^2$$

Assessment exit quiz

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6 Questions

Q1.
A right-angled triangle has a height of 12 cm and a hypotenuse of 15 cm. The area of the triangle is cm$$^2$$.
Correct Answer: 54
Q2.
An isosceles triangle has lengths 26 cm, 26 cm and 20 cm. The area of the isosceles triangle. is cm$$^2$$.
Correct Answer: 240
Q3.
A triangle is drawn. The adjacent lengths are 7 cm and 9 cm with a 47° angle in between. The area of the triangle, to 1 decimal place, is cm$$^2$$.
Correct Answer: 23.0
Q4.
An equilateral triangle has lengths of 6 cm. The area of the triangle, to 1 decimal place, is cm$$^2$$.
Correct Answer: 15.6
Q5.
Two adjacent lengths of a triangle are $$x$$ cm and 6 cm with a 47° angle in between. The area is 26.3 cm$$^2$$. Work out the value of $$x$$ , giving your answer rounded to the nearest integer.
Correct Answer: 12, 12 cm
Q6.
An equilateral triangle with lengths of $$x$$ cm is drawn. It has an area of 3.9 cm$$^2$$. Work out the length of the equilateral triangle, giving your answer as a whole number.
Correct Answer: 3 cm, 3

Lesson appears in

UnitMaths / Non right-angled trigonometry

Maths

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