Area from an included angle
Two sides are 6 and 8 with included angle 30°. Find the area.
- Use ½ × 6 × 8 × sin 30°.
- sin 30° = ½, so multiply 24 by ½.
Answer: 12 square units
Triangle area and side lengths
Match the known information to an area, cosine-rule or sine-rule calculation.
Before you start
You’ll need Pythagoras in three dimensions. Follow a link if you want to revise it first.
Using the known sides and angles
For two sides and the included angle, area = ab sin C / 2. The angle must lie between those sides.
The cosine rule finds the opposite side from two sides and their included angle. Pythagoras is its right-angle special case.
The sine rule can give two triangles from side-side-angle data. A valid acute answer need not be the only answer.
With sides 6 and 8, the area is 12 when the included angle is 30°, and 24 when it is 90°.
Area and the opposite side
Recover Pythagoras
Two sides are 3 and 4 with included angle 90°. Find the opposite side.
- Use c² = 3² + 4² − 2 × 3 × 4 × cos 90°.
- cos 90° = 0, so c² = 25; take the positive root for a length.
Answer: 5 units
A length is positive; an area formula also needs the factor ½.
Practice questions
The first link uses smaller values. Move on when you can answer without referring to the examples. The difficulty settings aren’t tied to exam grades.
- Triangle area from an included angle — smaller values
- Cosine rule with an exact side
- Sine rule with an exact squared side
Practice is filtered to your selected course. Change course or options when needed.
Check your understanding
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Why must you check for a second angle when using the inverse sine?