Area of a Triangle — The General Formula

The general area formula, Area = ½ab sin C, finds the area of any triangle from two sides and their included angle — no height needed.

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The familiar area rule, \tfrac{1}{2} \times \text{base} \times \text{height} , needs the height — and in a triangle that is not right-angled the height is rarely given. The general formula avoids it completely: two sides and the angle between them are enough.

Theorem Area from two sides and their angle

\text{Area} = \tfrac{1}{2}\,a\,b\,\sin C

C a b h
a and b: the two known sides.
C: the angle between them — the included angle.
Works for any triangle: acute, right or obtuse.

The height is hidden inside the sine: h = b\sin C , so \tfrac{1}{2}ah becomes \tfrac{1}{2}ab\sin C . Nothing new is being claimed — the same area, written without h .

Note The right angle is just a special case

When C = 90° , \sin C = 1 and the formula collapses to \tfrac{1}{2}ab — exactly \tfrac{1}{2} \times \text{base} \times \text{height} , because the two sides are then the base and the height. The general formula contains the old one.

Example Three applications
In triangle ABC: AB = 8 , AC = 6 , \angle A = 30° .
\text{Area} = \tfrac{1}{2}(8)(6)\sin 30° = 24 \times 0.5
⟹ Area = 12 cm²
In triangle XYZ: XY = 10 , XZ = 7 , \angle X = 120° (obtuse).
\text{Area} = \tfrac{1}{2}(10)(7)\sin 120° = 35 \times 0.8660
⟹ Area ≈ 30.31 cm²
With a = 5 and b = 8 fixed, \tfrac{1}{2}ab = 20 , so the area is 20\sin C :
at 45°: 20\sin 45° \approx 14.14  ·  at 90°: 20 \times 1 = 20  ·  at 135°: 20\sin 45° \approx 14.14
⟹ 45° and 135° give the same area; 90° gives the largest
Reference How the angle changes the area (a = 5, b = 4)
Angle C sin C Area
0° 0 0
30° 0.5 5
90° 1 (largest) 10 (maximum)
150° 0.5 5
180° 0 0

The area grows from 0° up to 90°, then shrinks again toward 180°. Supplementary angles give equal areas because \sin\theta = \sin(180° - \theta) .

Summary
  1. Area = ½ × a × b × sin C, where C is the angle included between sides a and b.
  2. It works for every triangle — no height needed.
  3. The area is largest at C = 90°, because sin 90° = 1.
  4. Supplementary included angles (θ and 180° − θ) give the same area.
231. Arithmetic Sequences and Series
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