Geometry and Trigonometry - Area of triangles

In triangle $ABC$, side $AC = 12\text{ cm}$, side $BC = 15\text{ cm}$, and angle $C = 35^\circ$. Calculate the area of the triangle.

$A = \frac{1}{2} \times 12 \times 15 \times \sin(35^\circ)$ $A = 90 \times 0.5735...$ $A \approx 51.6\text{ cm}^2$

Since we are given two sides and the included angle, we use the trigonometric area formula $A = \frac{1}{2}ab \sin C$.

Find the area of a triangle with side lengths $a = 5\text{ cm}$, $b = 6\text{ cm}$, and $c = 9\text{ cm}$.

First, find the semi-perimeter: $s = \frac{5 + 6 + 9}{2} = 10$ Use Heron's Formula: $A = \sqrt{10(10-5)(10-6)(10-9)}$ $A = \sqrt{10 \times 5 \times 4 \times 1}$ $A = \sqrt{200} \approx 14.1\text{ cm}^2$

When three sides are given, Heron's Formula is the most direct method. Calculate the semi-perimeter $s$ first, then substitute into the area formula.

A triangle has an area of $15\text{ cm}^2$. Two of its sides are $5\text{ cm}$ and $8\text{ cm}$. Find the possible values for the included angle $\theta$.

$15 = \frac{1}{2}(5)(8) \sin \theta$ $15 = 20 \sin \theta$ $\sin \theta = \frac{15}{20} = 0.75$ $\theta = \arcsin(0.75) \approx 48.6^\circ$ Or the obtuse possibility: $\theta = 180^\circ - 48.6^\circ = 131.4^\circ$

Rearrange the area formula to solve for $\sin \theta$. Since the sine of an angle is positive in both the first and second quadrants, there are two possible triangles unless otherwise specified.

1. In an equilateral triangle, all internal angles are $60^\circ$.
2. Identify the sides: $a = 8$ and $b = 8$.
3. Identify the included angle: $C = 60^\circ$.
4. Apply the formula $A = \frac{1}{2}ab \sin C$: $A = \frac{1}{2}(8)(8) \sin(60^\circ)$
5. Recall that $\sin(60^\circ) = \frac{\sqrt{3}}{2}$: $A = 32 \times \frac{\sqrt{3}}{2} = 16\sqrt{3}$
6. Final Area = $16\sqrt{3}\text{ cm}^2$.

Since all angles in an equilateral triangle are $60^\circ$, we can use any two sides and the $60^\circ$ angle between them in the sine area formula.

1. Use Heron's Formula. First, find the semi-perimeter $s$: $s = \frac{7 + 10 + 13}{2} = \frac{30}{2} = 15$
2. Substitute $s, a, b, c$ into the area formula: $A = \sqrt{15(15-7)(15-10)(15-13)}$ $A = \sqrt{15 \times 8 \times 5 \times 2}$
3. Calculate the product: $A = \sqrt{1200}$
4. Evaluate the square root: $A \approx 34.641...$
5. Area = $34.64\text{ m}^2$.

When three sides are given without angles, Heron's Formula is the most direct method to find the area.