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

The idea

This Higher-only result lets you find a triangle's area from two sides and the included angle, or work backwards to find a missing side or angle. The formula is Area = ½ab sin C, where a and b are the two sides meeting at angle C. For example, if a = 7 cm, b = 9 cm and C = 40°, then Area = ½ × 7 × 9 × sin 40° ≈ 20.2 cm². To find a side, substitute the known area, one side and the angle, then rearrange: 2 × Area ÷ (b sin C) = a. To find an angle, rearrange to sin C = 2 × Area ÷ (ab), then use the inverse sine. If the triangle is obtuse at C, subtract the calculator's acute answer from 180°, because sin(180° - θ) = sin θ. Your calculator must be in degree mode. This formula works for any triangle, extending your mensuration toolkit beyond ½ × base × height.

What earns the marks

• Selects the correct formula Area = ½ab sin C and identifies a and b as the two sides that include angle C.
• Substitutes known values correctly, including using the included angle rather than a non-included angle.
• Calculates the area accurately, for example ½ × 7 × 9 × sin 40° ≈ 20.2 cm², with correct units.
• Rearranges the formula correctly to find a missing side: a = 2 × Area ÷ (b sin C).
• Rearranges the formula correctly to find a missing angle: sin C = 2 × Area ÷ (ab), then applies inverse sine.
• Adjusts the angle for an obtuse triangle by subtracting the acute inverse sine value from 180°.
• Uses degree mode on the calculator when the angle is given in degrees and rounds appropriately.

Common mistakes and how to put them right

• Using a side that is not part of the included angle.

  Put it right: The formula needs the angle between the two sides being multiplied.

• Forgetting the factor of ½.

  Put it right: The area is half the product ab sin C, so omitting ½ doubles the answer.

• Rearranging incorrectly when finding a side or angle.

  Put it right: For a side, divide 2 × Area by (the other side × sin C).

• Forgetting to subtract the acute angle from 180° when finding an obtuse included angle, leading to an incorrect acute answer.

• Leaving the calculator in radian mode.

  Put it right: When the angle is in degrees, the calculator must be in degree mode.

Exam tips

• Label the triangle clearly before substituting: mark the two sides you will use and the angle between them as C.
• When finding an angle, check the diagram to see if it is obtuse; if so, remember to subtract your calculator's answer from 180°.
• Give units for area (for example cm²) and for lengths (for example cm), and round only at the final step unless the question says otherwise.