Exploring the Area of Triangles with Various Formulas and Examples

Triangles are a fundamental element in geometry, used in numerous applications from everyday tasks to complex engineering designs. Knowing how to calculate the area of a triangle is crucial for many fields, including architecture, construction, and graphic design. In this article, we will explore several methods for finding the area of a triangle, focusing on Heron's formula, the cosine rule, and some practical examples.

Heron's Formula for Finding the Area of a Triangle

Heron's formula is a powerful method to find the area of a triangle when the lengths of all three sides are known. The formula is as follows:

Area √[s (s - a) (s - b) (s - c)]

Where s is the semi-perimeter, given by:

s (a b c) / 2

Let's use this formula to find the area of a triangle with sides 8 cm, 11 cm, and 13 cm.

Example with Heron's Formula

Given:a  8 cm, b  11 cm, c  13 cmCalculate s (the semi-perimeter):s  (8   11   13) / 2  32 / 2  16 cmUsing Heron's formula:Area  √[16 (16 - 8) (16 - 11) (16 - 13)]  √[16 x 8 x 5 x 3]  4 x 2 x √30  8√30 cm2The area of the triangle is 43.82 cm2.

The Cosine Rule: Finding the Area Using the Cosine of an Angle

When two sides and the included angle of a triangle are known, the cosine rule can be used to find the third side, and then Heron's formula can be applied. Let's solve a triangle with sides 10 cm, 12 cm, and 14 cm using this method.

Example with the Cosine Rule

Given a triangle with sides a 10 cm, b 14 cm, c 16 cm, and using the cosine rule to find the area:

Using the cosine rule to find angle C:cos C  (102   142 - 162) / (2 x 10 x 14)  144 - 256 / 280  -112 / 280  -1/4Therefore, sin2C  1 - cos2C  1 - (1/4)2  1 - 1/16  15/16sin C  √15/4Area  10 x 16 x √15 / 8  20√15 cm2The area of the triangle is 20√15 cm2.

Alternative Methods and Exercises

A third method for finding the area of a triangle is demonstrated by Philip Lloyd, which involves an alternative formula:

Area of triangle 1/4 √(a2b2 b2c2 c2a2 - a? - b? - c?)

Using this formula for a triangle with sides a 12 cm, b 14 cm, and c 16 cm:

Area  1/4 √(122 x 142   142 x 162   162 x 122 - 12? - 14? - 16?)      1/4 √(20736   32256   27648 - 20736 - 32256 - 65536)      1/4 √15360      1/4 x 39  9.75The area of the triangle is 9.75 cm2.

These examples demonstrate the versatility of different formulas for finding the area of a triangle. Whether you need to use the semi-perimeter method, the cosine rule, or an alternative formula, understanding these methods enhances your problem-solving skills in geometry.