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

Starter quiz

Match the areas of these triangles with their dimensions.

Area = 15 cm $^2$

⇔

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

Area = 22 cm $^2$

⇔

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

Area = 16 cm $^2$

⇔

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

Area = 4.5 cm $^2$

⇔

A triangle with perpendicular height = 3 cm and base = 3 cm ✓
Which of the following calculate the area of this triangle?

$\frac{1}{2}\times 6.3\times5\times\text{sin}(55.3°)$ ✓

$\frac{1}{2}\times 6.3\times5.4\times\text{sin}(49.8°)$ ✓

$\frac{1}{2}\times 5.4\times5\times\text{sin}(74.9°)$ ✓

$\frac{1}{2}\times 6.3\times5.4\times\text{sin}(55.3°)$

$\frac{1}{2}\times 5.4\times5\times\text{sin}(55.3°)$
The area of this triangle, given to 2 significant figures, is ______cm $^2$ .

'13 ' ✓
A right angled triangle has a height of 12 cm and a hypotenuse of 13 cm. Work out the area.

30 cm $^2$ ✓

60 cm $^2$

15 cm $^2$

120 cm $^2$
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.

$13.5\sqrt{3}$ cm $^2$ ✓

$27\sqrt{3}$ cm $^2$

$\sqrt{3}$ cm $^2$

$27\sqrt{3}$ cm $^2$
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 $^2$

10 cm $^2$ ✓

20 cm $^2$

40 cm $^2$

Exit quiz

A right-angled triangle has a height of 12 cm and a hypotenuse of 15 cm. The area of the triangle is ______cm $^2$ .

'54' ✓
An isosceles triangle has lengths 26 cm, 26 cm and 20 cm. The area of the isosceles triangle. is ______cm $^2$ .

'240' ✓
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 $^2$ .

'23.0' ✓
An equilateral triangle has lengths of 6 cm. The area of the triangle, to 1 decimal place, is ______cm $^2$ .

'15.6' ✓
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.

'12' ✓
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.

'3 cm' ✓

Worksheet

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Video

Lesson Details

Key learning points

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

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.

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.