> ## Content Index
> Fetch the complete content index at: https://www.esheets.io/llms.txt
> Use this file to discover other available public pages before exploring further.

# Area of a Triangle Using Sine
- URL: https://www.esheets.io/area-of-a-triangle-using-sine/
- Published: 2026-10-24T19:34:32.000Z
- Updated: 2026-10-24T19:49:37.000Z
- Author: Richard Linnington
- Tags: Maths, Geometry and Shape

Practise calculating the area of a triangle from two sides and their included angle using **A = ½ab sin C**. The questions also develop rearranging the formula to find a missing side, choosing the correct acute or obtuse angle, and solving an algebraic area problem.

Use a calculator in degree mode and pay close attention to the requested rounding. [Jump to the questions](#practise-now)

## Practise now

Worksheet preview and key skills 

### Worksheet preview

Eight self-marking questions use clear triangle diagrams to practise the formula A = ½ab sin C.

The sequence moves from finding areas to missing sides, acute and obtuse included angles, and a final algebraic problem.

#### What you’ll practise

- Finding triangle areas to a stated accuracy.
- Rearranging the formula to find a side.
- Using the correct inverse-sine branch for an acute or obtuse angle.

Use the interactive worksheet below, or read the [Topic guide](#topic-guide) for the method and worked example.

Use **A = ½ab sin C**, where C is the included angle. Use degree mode and give each answer to the stated accuracy.

## Topic guide

### The formula

When two side lengths and the angle between them are known, the area of a triangle is **A = ½ab sin C**. The angle C must be the included angle: the angle where sides a and b meet. Set your calculator to degree mode.

### Finding an area

For sides 7 cm and 10 cm with an included angle of 50°, A = ½ × 7 × 10 × sin 50° = 26.8 cm² to 1 decimal place.

### Finding a missing side

Rearrange A = ½ab sin C to make a side the subject. For example, **a = 2A ÷ (b sin C)**. Keep the full calculator value until the final rounding step.

### Finding the included angle

First calculate sin C = 2A ÷ (ab), then use inverse sine. A positive sine value can correspond to an acute angle θ or the supplementary obtuse angle 180° − θ. Use the statement that the angle is acute or obtuse to choose the correct branch.

### Algebra example

If the enclosing sides are (x + 3) cm and (x + 5) cm, the angle is 30°, and the area is 12 cm², then ½(x + 3)(x + 5)sin 30° = 12\. Since sin 30° = ½, (x + 3)(x + 5) = 48.

This simplifies to x² + 8x − 33 = 0, so (x − 3)(x + 11) = 0\. The roots are 3 and −11, but −11 would give negative side lengths, so **x = 3**.

### Common mistakes

- Using an angle that is not between the two known sides.
- Leaving the calculator in radians instead of degrees.
- Rounding during an intermediate calculation.
- Forgetting the supplementary obtuse angle when using inverse sine.