Calculating Area of Triangles

Study Notes

Area of a Triangle

Triangles are fundamental shapes in geometry, with properties that are well understood. They have many applications ranging from construction blueprints to computer graphics and physics simulations. Knowing how to calculate the area of different kinds of triangles is important for solving problems involving these angular polygons. Let's explore the formulas for calculating the area of various types of triangles.

Formula

The basic formula for calculating the area of a triangle is derived using the determinant:

Area = 1/2 * base * height

However, this formula assumes that you already know two sides of the triangle, which is usually not the case when dealing with unknown triangles. To find the area without knowing two of its sides, we can use Heron's formula.

Heron's Formula

This simple algorithm allows us to compute the semiperimeter s, first by computing it directly by adding the lengths of all three edges a, b, and c. It then computes the area A using the following formula:

A^2 = s(s - a)(s - b)(s - c)

Taking the square root of both sides gives a quadratic in A, which can be solved by completing the square. This produces the final formula:

A = sqrt(s(s - a)(s - b)(s - c))

Heron's formula can be used to calculate the area of a triangle using the lengths of its three sides, without knowing the height.

Types of Triangles

There are several types of triangles, and they can be classified based on their sides or angles.

Scalene Triangle

A scalene triangle is one in which all three sides are of different lengths. In a scalene triangle, one side is always greater than the others. It is the most common type of triangle.

Isosceles Triangle

An isosceles triangle has two sides of equal length. An isosceles triangle has two angles of equal measure. The third angle is always greater than 60 degrees.

Equilateral Triangle

An equilateral triangle has three sides of equal length. All angles in an equilateral triangle are also equal, each measuring 60 degrees. These triangles have equal angles and equal sides.

Right Triangle

A right triangle is a triangle in which one of the internal angles is a right angle (90 degrees). The sides opposite the right angle are called the hypotenuse and the legs. The hypotenuse is the longest side of the triangle.

Area of Special Triangles

Area of an Equilateral Triangle

An equilateral triangle has three sides of equal length, and all angles are equal to 60 degrees. Therefore, the area of an equilateral triangle is:

Area = (sqrt(3)/4) * side^2

Area of a Right Triangle

Area = (base * height) / 2

This formula works because in a right triangle, the area is simply the base times the height divided by 2.

Description

Explore the different formulas for calculating the area of triangles including the classic 1/2 * base * height formula and Heron's formula to find the area without knowing two sides. Learn how to calculate the area of special triangles like equilateral triangles and right triangles. Understand the properties of scalene, isosceles, and equilateral triangles.