Solution of Triangles: Area of Triangle, Heron's Formula

The solution of triangles is an important topic in trigonometry that deals with finding various properties of a triangle, such as its sides, angles, area, and other dimensions, given some initial information. One of the key aspects of solving triangles is determining the area of a triangle. There are several formulas to calculate the area, depending on the given information. In this content, we will focus on the area of a triangle and Heron's formula.

Area of a Triangle

The area of a triangle can be calculated in various ways, depending on the known elements of the triangle. Here are some common formulas:

1. Base and Height: If the base ( b ) and the height ( h ) of a triangle are known, the area ( A ) can be calculated using the formula: [ A = \frac{1}{2} \times b \times h ]

2. Two Sides and Included Angle: If two sides ( a ) and ( b ) and the included angle ( C ) are known, the area can be calculated using the formula: [ A = \frac{1}{2} \times a \times b \times \sin(C) ]

3. Using Trigonometry in Right Triangles: In a right-angled triangle, if the lengths of the two legs (perpendicular sides) are known, the area is simply half the product of those two sides.

4. Using Coordinates: If the vertices of the triangle are given by their coordinates in the Cartesian plane, the area can be found using the determinant method.

Heron's Formula

Heron's formula is a special case used to find the area of a triangle when all three sides are known. It is named after Hero of Alexandria, a Greek engineer and mathematician.

The formula states that for a triangle with sides of lengths ( a ), ( b ), and ( c ), the area ( A ) can be found using the semi-perimeter ( s ), which is half the perimeter of the triangle:

[ s = \frac{a + b + c}{2} ]

Then, the area ( A ) is given by:

[ A = \sqrt{s(s - a)(s - b)(s - c)} ]

Important Points about Heron's Formula

• It requires no knowledge of the angles of the triangle.
• It is applicable to any kind of triangle, whether it is acute, obtuse, or right-angled.
• It is especially useful in situations where the height of the triangle is difficult to determine.

Table: Comparison of Area Formulas

Formula Type	Requirements	Applicability	Complexity
Base and Height	Base length and height	Any triangle	Simple
Two Sides and Included Angle	Two sides and the angle between them	Any triangle	Moderate
Right Triangle Trigonometry	Lengths of the two legs	Right-angled triangles only	Simple
Coordinates	Coordinates of the vertices	Any triangle in a plane	Moderate
Heron's Formula	Lengths of all three sides	Any triangle	Moderate

Examples

Example 1: Base and Height

Given a triangle with a base of 6 units and a height of 4 units, find the area.

Using the base and height formula:

[ A = \frac{1}{2} \times 6 \times 4 = 12 \text{ square units} ]

Example 2: Heron's Formula

Consider a triangle with sides of lengths 7 units, 8 units, and 5 units. Find the area using Heron's formula.

First, calculate the semi-perimeter:

[ s = \frac{7 + 8 + 5}{2} = 10 \text{ units} ]

Now, apply Heron's formula:

[ A = \sqrt{10(10 - 7)(10 - 8)(10 - 5)} ] [ A = \sqrt{10 \times 3 \times 2 \times 5} ] [ A = \sqrt{300} ] [ A = 10\sqrt{3} \text{ square units} ]

Example 3: Two Sides and Included Angle

Given a triangle with sides of lengths 5 units and 7 units, and an included angle of 60 degrees, find the area.

Using the two sides and included angle formula:

[ A = \frac{1}{2} \times 5 \times 7 \times \sin(60^\circ) ] [ A = \frac{1}{2} \times 5 \times 7 \times \frac{\sqrt{3}}{2} ] [ A = \frac{35\sqrt{3}}{4} \text{ square units} ]

By understanding these formulas and methods, one can efficiently solve for the area of a triangle in various scenarios, which is a fundamental skill in geometry and trigonometry.