Geometry and measures - Mensuration and calculation: know and apply Area = ½ab sin C to calculate the area, sides or angles of any triangle (Higher tier only)

The idea

This is Higher tier only. The formula Area = ½ab sin C gives the area of any triangle from two sides and the angle between them, with C the angle sitting between the sides a and b. That angle has to be the included one; if the angle you are given lies somewhere else, find the included angle first. For sides of 8 cm and 5 cm with an angle of 30° between them, the area is ½ × 8 × 5 × sin 30° = 10 cm². The formula rearranges as well: given the area and the two sides, sin C = (2 × Area)/(ab), and the inverse sine gives the angle, although an obtuse angle can fit the same sine. Longer questions pair it with the cosine rule, or subtract this triangle from a sector to leave the area of a segment.

What earns the marks

• A correct substitution with the included angle in place, such as half of eight times five times the sine of the angle.
• The area evaluated and written in square units.
• A correct rearrangement when the area is given and a side or the angle is what is wanted.
• Finding the missing angle or side first, by the sine or cosine rule, when the included angle is not handed to you.

Common mistakes and how to put them right

• Dropping the half, so the area comes out twice the size it should be.

• Using an angle that does not lie between the two sides named in the formula.

• Giving the answer in centimetres rather than square centimetres.

• Putting a perpendicular height into the formula as one of the two sides.

Exam tips

• Check that the angle you are about to use lies between the two sides you have; if it does not, the formula is not ready yet.
• Make sure the calculator is set to degrees before you press the sine key.
• For a segment, work out the sector area and the triangle area separately and write both down before you subtract.