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.

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

Area $= \frac{1}{2}ab\sin C$ is the familiar formula in disguise. Area is

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

. 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.

The angle must be the one between the two sides you use — the included angle. It's the same "in-between" requirement as the cosine rule and SAS congruence, and for the same reason: two sides and the angle between them fix the triangle completely, so they determine its area.

In reverse, it finds an angle. Given the area and two sides, rearrange to

\sin C = \frac{2 \times \text{Area}}{ab}

, then

\sin^{-1}

. As always with sine, there may be an obtuse partner

180° - C

that gives the same area — so a question wanting one answer will tell you the angle is acute or obtuse.

3D: the whole method is "find the right-angled triangle". A 3D question is a 2D question hiding inside a solid. The routine:

Sketch the triangle you need on its own, flat on the page, away from

the solid.

Fill in the lengths you know, working out any that need Pythagoras

across a face first — typically a base diagonal.

Then it's ordinary trigonometry: SOHCAHTOA if the triangle is

right-angled, sine/cosine rule if not.

The angle between a line and a plane is the angle between the line and its shadow on that plane — drop a perpendicular from the top of the line to the plane, join the foot to where the line meets the plane, and that right-angled triangle contains the angle you want.

Don't round the intermediate. 3D answers usually need a length found in step 2 before the trig in step 3; keep it in the calculator (or keep it squared) rather than rounding, or the final answer drifts off the mark scheme.