Area of a Triangle Using Trigonometry – for GCSE Maths
This blog explains what is the area of a triangle formula, when it should be used, and how to apply it confidently to exam-style questions involving non-right-angled triangles.
You will learn how to use the formula to calculate the area of a triangle when the vertical height is not given, as well as how this method can be extended to solve more advanced problems, including finding missing sides, missing angles, and areas of other polygons.
Look out for our Area of a Triangle worksheets and GCSE-style exam questions at the end to practise and consolidate what you have learned.
We also provide GCSE-style maths assessment for your child to take for free
Page Contents
What is the Area of a Triangle of Trigonometry?
When students first learn how to find the area of a triangle, they are usually taught the familiar formula:
This method works perfectly when the vertical height is known.
However, in many GCSE exam questions, the height is not given directly. Instead, we may be given side lengths and angles.
This is where trigonometry provides a powerful alternative. Using trigonometry, we can calculate the area of any triangle, even when the perpendicular height is unknown.
In this blog, we explore how to find the area of a triangle using trigonometry, focusing on the key formula:
To use this formula, we need:
- The lengths of at least two sides of the triangle
- The included angle between those two sides
This situation occurs frequently in GCSE questions involving non-right-angled triangles.
Comparing the Two Area of a Triangle Formulas
Example 1 — Using Base and Height
Suppose a triangle has:
- Base = \( 10 \, \text{cm} \), Height = \( 6 \, \text{cm} \)
Then, the area is calculated as:
\text{Area} = \frac{1}{2} \times 10 \times 6 = 30 \, \text{cm}^2
\]
This works because the height is perpendicular to the base.
Example 2 — Using Trigonometry for area of a triangle
Now suppose we know:
- Side \( a = 8 \, \text{cm} \), Side \( b = 11 \, \text{cm} \), Angle \( C = 35^\circ \)
The height is not given, but we can still find the area:
\text{Area} = \frac{1}{2} \times 8 \times 11 \times \sin 35^\circ
\]
\text{Area} \approx 25.2 \, \text{cm}^2
\]
Worked Examples
1. Two Sides and the Included Angle
Given: $a = 9\text{ cm}$, $b = 12\text{ cm}$, $C = 50^\circ$
Calculate the area of the triangle $ABC$. Write your answer to $3$ significant figures.
$$\text{Area} = \frac{1}{2}ab\sin C$$$$A=\frac{1}{2} \times 9 \times 12 \times \sin 50^\circ \approx 41.3\text{ cm}^2$$
2. Three Sides and One Angle
Calculate the area of the scalene triangle $PQR$. Write your answer to $3$ significant figures.
$$A=\frac{1}{2} \times 7 \times 10 \times \sin 62^\circ \approx 30.9\text{ cm}^2$$
3. Isosceles Triangle with a Known Angle
Triangle $XYZ$ is an isosceles triangle. Find the area of the triangle to $3$ significant figures.
As the triangle $XYZ$ is isosceles, $XY=XZ=10 \text{ cm}$
$$A=\frac{1}{2} \times 10 \times 18 \times \sin 35^\circ \approx 51.6\text{ cm}^2$$
4. Finding a Length Given the Area
$$\text{Area} = \frac{1}{2}ab\sin C$$
$$24 = \frac{1}{2} \times a \times 8 \times \sin 45^\circ$$
$$a \approx 8.49\text{ cm}$$
5. Finding an Angle Given the Area
The area of this triangle is $30\text{ cm}^2$. Find the angle $C$.
$$30 = \frac{1}{2} \times 9 \times 10 \times \sin C$$
$$\sin C = \frac{2}{3},\quad C \approx 41.8^\circ$$
Where Does the Formula Come From?
We derive the formula for the area of any triangle by taking the triangle $ABC$.
If we drop a perpendicular from one vertex of the triangle, the height can be expressed using trigonometry:
Substituting this into the standard area formula:
$$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$
$$\text{Area} = \frac{1}{2} \times a \times (b\sin C) = \frac{1}{2}ab\sin C$$
This shows that the trigonometric area formula is simply an extension of the original base–height method.
Common Misconceptions
- Using two sides that do not enclose the given angle
- Applying the sine of the wrong angle
- Assuming the triangle is right-angled
- Using inverse sine instead of sine
- Rounding values too early
We also provide area of triangle worksheets designed around GCSE-style exam questions to help students practise applying this formula accurately, contact us to get for free!
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