The Area of a Triangle via Trigonometry — Mathematics Knowledge System

Overview

The standard area formula for a triangle, (1/2)·base·height, requires knowing an altitude, which is often inconvenient to measure directly. The trigonometric area formula replaces the height with a side length and the sine of the included angle, so that the area of any triangle can be computed from two sides and the angle between them (SAS data) without ever constructing an altitude. This is exactly the same formula that underlies the cross-product magnitude formula for the area of a parallelogram in vector geometry, and it combines naturally with the Law of Sines and Law of Cosines to solve for areas of triangles specified in any of the standard ways (SSS, SAS, ASA).

Intuition

Drop an altitude h from the vertex between sides a and b down to side a (or its extension), splitting the triangle so that h relates to side b and angle C by right-triangle trigonometry: sinC = h/b, so h = b sinC. Substituting this into the ordinary area formula (1/2)·base·height with base = a gives Area = (1/2)a(b sinC) = (1/2)ab sinC. Because the same triangle can be described starting from any of its three vertices, the same reasoning with a different base/altitude pair yields the two other equivalent forms — all three describe the identical area, just built from a different pair of sides and their included angle.

Formal Definition

Definition

For a triangle with sides a, b, c opposite angles A, B, C respectively, the area using two sides and the included angle is:

\text{Area} = \frac{1}{2}ab\sin C = \frac{1}{2}bc\sin A = \frac{1}{2}ac\sin B

SAS area formula

h = b\sin C \quad \text{(altitude from side } a\text{)}

Height in terms of side and angle

Worked Examples

Apply Area = (1/2)ab sinC.
$\text{Area} = \frac{1}{2}(7)(10)\sin(40°)$
sin(40°) ≈ 0.6428, so compute the product.
$= 35 \times 0.6428 \approx 22.50$

Answer: Area ≈ 22.50 square units

Practice Problems

Difficulty 3/10

Find the area of a triangle with sides a = 12, b = 15, and included angle C = 30°.

Difficulty 4/10

A triangular plot has two sides of length 40 m and 25 m meeting at a 60° angle. Find its area.

Difficulty 6/10

A triangle has sides a = 7, b = 10, and included angle C = 40°. Using the Law of Cosines, find side c, then verify Heron's formula gives the same area as the trigonometric formula (1/2)ab sinC ≈ 22.50.

Quiz

Summary

Area = (1/2)ab sinC uses two sides and their included angle, avoiding the need to measure an altitude directly.
It is derived by expressing the triangle's altitude as (side)×sin(included angle) and substituting into (1/2)·base·height.
It agrees with Heron's formula and the Law of Cosines/Sines, giving multiple consistent ways to compute a triangle's area.

References

WebsiteWikipedia — Solution of triangles
WebsiteWikipedia — Trigonometric functions