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

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

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

Maths

In the prior unit, pupils explored the trigonometric ratios for right-angled triangles in a variety of situations. In this unit, pupils explore what can be done when we have a triangle that is not right-angled.

Non right-angled trigonometry

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

An isosceles triangle has lengths 26 cm, 26 cm and 20 cm. The area of the isosceles triangle. is {{}}cm$$^2$$.

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$$^2$$.

An equilateral triangle has lengths of 6 cm. The area of the triangle, to 1 decimal place, is {{}}cm$$^2$$.

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. 

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. 

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

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

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

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

Match the areas of these triangles with their dimensions. 

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

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

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

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

$$\frac{1}{2}\times 5.4\times5\times\text{sin}(55.3°)$$

Which of the following calculate the area of this triangle?

The area of this triangle, given to 2 significant figures, is {{}}cm$$^2$$.

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

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. 

An isosceles triangle has a base of 4 cm and base angles of 68.2°. Select the correct area given to 2 significant figures. 

This lesson will be removed by end of Summer Term 2025.

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

Lesson outcome

This lesson will be removed by end of Summer Term 2025.