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Area of a Triangle using Trigonometry

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Right triangle with one angle of 30°, and the adjacent side to this angle is 8 units long.

\text{Area} = \frac{1}{2}ab\sin(C) = \frac{1}{2} \times 8 \times 16 \times \sin(90^\circ) = 64\ \text{units}^2

An isosceles triangle with two angles of 45° and one side of length 1 unit along the angle of 90°.

\text{Area} = \frac{1}{2}ab\sin(C) = \frac{1}{2} \times 1 \times 1 \times \sin(90^\circ) = 0.5\ \text{units}^2

Isosceles triangle with base angles of 70°, vertex angle of 40°, and base of 10 units.

\text{Area} = \frac{1}{2}ab\sin(C) = \frac{1}{2} \times 10 \times 10 \times \sin(40^\circ)\ \text{(where $a$ is the altitude)}

Triangle with angles 45°, 45°, 90° and one leg of 7 units.

\text{Area} = \frac{1}{2}ab\sin(C) = \frac{1}{2} \times 7 \times 7 \times \sin(90^\circ) = 24.5\ \text{units}^2

Triangle with angles of 30°, 60°, and 90°, with the longer leg measuring 9 units.

\text{Area} = \frac{1}{2}ab\sin(C) = \frac{1}{2} \times 9 \times 9\sqrt{3} \times \sin(90^\circ) = \frac{81\sqrt{3}}{2}\ \text{units}^2

Right triangle with hypotenuse of 13 units and one angle of 45°.

\text{Area} = \frac{1}{2}ab\sin(C) = \frac{1}{2} \times 9.192 \times 9.192 \times \sin(90^\circ) = 42.125\ \text{units}^2

Equilateral triangle with each side 6 units.

\text{Area} = \frac{1}{2}ab\sin(C) = \frac{1}{2} \times 6 \times 6 \times \sin(60^\circ) = 9\sqrt{3}\ \text{units}^2

Triangle with sides of 5 units, 12 units and an included angle of 120°.

\text{Area} = \frac{1}{2}ab\sin(C) = \frac{1}{2} \times 5 \times 12 \times \sin(120^\circ) = 15\sqrt{3}\ \text{units}^2

Scalene triangle with sides 7 units, 8 units, and an included angle of 60° between them.

\text{Area} = \frac{1}{2}ab\sin(C) = \frac{1}{2} \times 7 \times 8 \times \sin(60^\circ) = 14\sqrt{3}\ \text{units}^2

Triangle with angles 30°, 60°, 90° and a hypotenuse of length 10 units.

\text{Area} = \frac{1}{2}ab\sin(C) = \frac{1}{2} \times 5 \times 10 \times \sin(90^\circ) = 25\ \text{units}^2

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