Proving the Area of a Triangle Formula: 1/2bc SinA 1/2ab SinC 1/2ac SinB

In geometry, understanding how to calculate the area of a triangle is fundamental. One of the most important and versatile formulas is the area of a triangle given by:

Understanding the Triangle Area Formula

The area of triangle (ABC) can be calculated using the following formula:

[text{Area} frac{1}{2}bc sin A frac{1}{2}ab sin C frac{1}{2}ac sin B]

where (a), (b), and (c) are the lengths of the sides opposite angles (A), (B), and (C) respectively.

This formula is incredibly useful because it allows us to find the area of a triangle without knowing its height, provided we know the lengths of two sides and the included angle. In the following sections, we will provide a detailed proof of this formula.

Proof of the Area Formula

Step 1: Using Side (b) as Base

Consider triangle (ABC). Let (a), (b), and (c) be the lengths of the sides opposite angles (A), (B), and (C) respectively.

Our goal is to express the area of triangle (ABC) in terms of side lengths and the sine of one of the angles. We start by considering side (b) as the base.

The height (h) from vertex (A) to side (b) can be written as:

[h a sin B]

Now we can calculate the area:

[text{Area} frac{1}{2} times b times h frac{1}{2} times b times a sin B frac{1}{2}ab sin B]

Step 2: Using Side (a) as Base

Next, we consider side (a) as the base of the triangle.

The height (h) from vertex (B) to side (a) is:

[h c sin A]

Thus, the area is:

[text{Area} frac{1}{2} times a times h frac{1}{2} times a times c sin A frac{1}{2}ac sin A]

Step 3: Using Side (c) as Base

Finally, we take side (c) as the base of the triangle.

The height (h) from vertex (C) to side (c) is:

[h b sin C]

The area is then:

[text{Area} frac{1}{2} times c times h frac{1}{2} times c times b sin C frac{1}{2}bc sin C]

Conclusion

We have shown that:

[text{Area} frac{1}{2}ab sin C frac{1}{2}ac sin A frac{1}{2}bc sin B]

This means that the area of triangle (ABC) can be expressed in terms of any two sides and the sine of the included angle. This completes our proof.

Visual Representation

In the diagram, (a, b,) and (c) are the sides of triangle (ABC). The heights (AF, BD,) and (CE) are perpendicular to the respective bases (a, b,) and (c). The sine rule tells us:

[sin A frac{CE}{b}, sin B frac{AF}{c}, sin C frac{BD}{a}]

This gives us:

[CE frac{b sin A}{b}, AF frac{c sin B}{c}, BD frac{a sin C}{a}]

Therefore:

[text{Area} frac{1}{2}bc sin A frac{1}{2}ac sin B frac{1}{2}ab sin C]

Hence, the formula is proven.