The idea

Area of a triangle & 3D trigonometry

Why ½ab sin C works for any triangle, using it in reverse to find an angle, and the method for 3D problems — find the right-angled triangle inside the solid, then work in two dimensions.

A full journey — read it, play with it, work it, then earn real exam marks. Everything stays on the timeline below.

In this lesson — start anywhere

Geometry & measures · Trigonometry (sine & cosine rules)

Area of a triangle & 3D trigonometry

Why ½ab sin C works for any triangle, using it in reverse to find an angle, and the method for 3D problems — find the right-angled triangle inside the solid, then work in two dimensions.

Why it works

The familiar area formula in disguise

This lesson holds the last two tricks of GCSE trigonometry: an area formula that needs no height, and the move that turns a frightening 3D solid into an ordinary flat triangle. First the area. You know

\text{Area} = \frac{1}{2} \times \text{base} \times \text{perpendicular height}

— and

\frac{1}{2}ab\sin C

is that same formula in disguise. Take

b

as the base; the height from the far vertex is

a\sin C

, because the side

a

leans away from the base at angle

C

. So

\text{Area} = \tfrac{1}{2}ab\sin C.

Keep reading — free

The rest of the explanation, plus 2 worked examples you step through move by move.

Start free

Takes a minute — no card.