Area using ½ab sin C

Objective. Find triangle areas and reverse the formula.

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Learn: the key idea

The area formula ½ab sin C finds a triangle's area using two sides and the included angle between them, without needing the perpendicular height. Multiply half the product of the two known sides by the sine of the angle between them; the formula can also be rearranged to find a missing side or angle if the area is given instead. For sides 6 cm and 10 cm with an included angle of 50°, area = ½(6)(10)sin50° ≈ 22.98 cm².

Worked example

A triangle has two sides of length 8 cm and 12 cm with an included angle of 35°. Find its area using ½ab sin C.

1 $Write the formula: Area = ½ \times a \times b \times sin C$
2 $Substitute values: Area = ½ \times 8 \times 12 \times sin35°$
3 $Calculate sin35° \approx 0.5736, so Area = 48 \times 0.5736$
4 $Simplify: Area \approx 27.5 cm²$

Answer: ≈ 27.5 cm²

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Common mistake. Students sometimes use an angle that is not the included angle between the two chosen sides, giving an incorrect area. Always check the angle lies directly between the two side lengths being multiplied.

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