How to Find the Area of a Triangle Given Its Sides and Angles

When dealing with a triangle, it is essential to have accurate methods to calculate its area based on the given information. Whether you know the lengths of the sides, the angles, or a combination of both, several formulas can be employed to find the area. In this article, we will explore the most common methods to calculate the area of a triangle, including the Sine Formula, Heron's Formula, and the Angle-Angle-Side (AAS) or Angle-Side-Angle (ASA) methods. We'll also provide examples to illustrate each method.

Using the Sine Formula (SAS)

The Sine Formula is particularly useful when you know two sides and the included angle of the triangle (Side-Angle-Side, SAS). The formula for the area in this case is:

Area 1/2 * a * b * sin(C)

Where a and b are the lengths of the two sides, and C is the included angle between those two sides.

Example

Suppose you have a triangle with sides a 5 and b 7, and the included angle C 60°. To find the area, follow these steps:

Plug the values into the Sine Formula:

Area 1/2 * 5 * 7 * sin(60°)

Note that sin(60°) √3/2, so the equation becomes:

Area 1/2 * 5 * 7 * (√3/2) 35 * (√3/4) 35√3/4 ≈ 15.2

This calculation provides the area of the triangle using the given sides and included angle.

Using Heron's Formula (SSS)

When you know the lengths of all three sides of the triangle (Side-Side-Side, SSS), Heron's Formula can be employed to find the area. Before applying the formula, you need to calculate the semi-perimeter:

s (a b c) / 2

Once you have the semi-perimeter, you can use it to find the area of the triangle with the following formula:

Area √s(s - a)(s - b)(s - c)

Where s is the semi-perimeter, and a, b, and c are the lengths of the sides of the triangle.

Example

Consider a triangle with sides a 6, b 8, and c 10. To find the area using Heron's Formula:

Calculate the semi-perimeter:

s (6 8 10) / 2 12

Apply the formula for the area:

Area √12(12 - 6)(12 - 8)(12 - 10) √12 * 6 * 4 * 2 √576 24

Thus, the area of the triangle is 24 square units.

Using the Angle-Angle-Side (AAS) or Angle-Side-Angle (ASA) Methods

When you are given two angles and one side of the triangle (AAS or ASA), you can use trigonometry to find the lengths of the other sides and then apply the Sine Formula to determine the area. The key step here is to use the Law of Sines:

sin(A)/a sin(B)/b sin(C)/c

Once you have the lengths of all three sides, you can use the Sine Formula as previously explained.

To illustrate, let's assume you have a triangle with angles A 50°, B 60°, and side a 10. The included angle C 180° - 50° - 60° 70° (since the sum of angles in a triangle is 180°).

Apply the Law of Sines to find sides b and c:

sin(50°)/10 sin(60°)/b

b (sin(60°) * 10) / sin(50°) ≈ 12.31

sin(50°)/10 sin(70°)/c

c (sin(70°) * 10) / sin(50°) ≈ 12.71

Now use the Sine Formula to calculate the area:

Area 1/2 * 10 * 12.31 * sin(70°) ≈ 61.65

A quick verification of sin(70°) ≈ 0.94

Area ≈ 1/2 * 10 * 12.31 * 0.94 ≈ 58.34

Conclusion

To summarize, finding the area of a triangle involves different formulas based on the given information. The Sine Formula is ideal when you know two sides and the included angle (SAS). When you have all three side lengths (SSS), Heron's Formula can be used. Lastly, if you have two angles and one side (AAS or ASA), you can determine the area with the Sine Formula after using the Law of Sines to find the unknown sides. Each method provides a way to accurately calculate the area of a triangle, depending on the available information.