Area of a triangle: ½ab sin C
A farmer knows two sides of a triangular plot of land and the angle between them, but not its height. Change the sides and the angle and find the area with ½ab sin C.
Core concept
With two sides a and b and the angle C between them, the height on a is b sin C, so the area of the triangle is ½ab sin C
Topic: Trigonometry
The area: How much flat space a shape covers, measured in square units like cm². All key words of a triangle is ½ × base × height, but in a field the height is hard to measure. If you know two sides, a and b, and the angle: The amount of turn between two lines that meet, measured in degrees (°). All key words C between them, trigonometry gives the height: from the top corner drop a perpendicular onto side a; it is b sin C long. So the area is ½ × a × b sin C, written ½ab sin C. The largest area for the same two sides comes at C = 90°, when sin C = 1. Because sin(180° − C) = sin C, angles of 60° and 120° give the same area, and so do 30° and 150°.
What you will learn
- Find the area of a triangle as ½ab sin C.
- Explain why the height is b sin C.
- Notice that sin C and sin(180° − C) give the same area.
Before you start
- Know the sine ratio and the area of a triangle as ½ × base × height.
Materials
- A measuring tape and a protractor (or a compass app)
Equipment
- A calculator with sin
Procedure
- Record a = 8 m, b = 6 m and C = 60°.
- Make C = 90°, then 120°, then 150°, and record each.
- Record a = 6 m, b = 5 m and C = 30°.
How this simulation is simplified
- The plot is a flat triangle; the sides are measured in metres.