How to Calculate the Perimeter of a Triangle: A Comprehensive Guide

To calculate the perimeter of a triangle, sum the lengths of its sides. Use the Pythagorean theorem for right triangles and the distance formula for coordinates.

Basic Formula

The perimeter of a triangle is the sum of the lengths of its three sides. This is the most straightforward method to calculate the perimeter. If you know the lengths of all three sides, simply add them together to get the perimeter. For example, if a triangle has sides of 7, 8, and 16 units, the perimeter is calculated as:
$P = 7 + 8 + 16 = 31 \text{ units}$
This method applies to any type of triangle, whether it's equilateral, isosceles, or scalene.

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Right Triangles

In the case of a right triangle, you can use the Pythagorean theorem to find the length of the hypotenuse if only two sides are known. The theorem states:
$a^2 + b^2 = c^2$
where $a$ and $b$ are the lengths of the legs, and $c$ is the hypotenuse. Once you have all three sides, you can calculate the perimeter by summing them up. This method is particularly useful when dealing with right-angled triangles where one angle is 90 degrees.

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Coordinates Method

When the coordinates of the triangle's vertices are given, you can use the distance formula to find the lengths of the sides. The distance formula is:
$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
By calculating the distances between each pair of vertices, you can determine the lengths of the sides. Once you have these lengths, sum them to find the perimeter. This method is useful in coordinate geometry problems.

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Scalene Triangle

A scalene triangle has three sides of different lengths. To find the perimeter, you still use the basic formula of summing the lengths of all three sides. For example, if the sides are 3, 4, and 5 units, the perimeter is:
$P = 3 + 4 + 5 = 12 \text{ units}$
This method applies to any triangle with unequal sides, ensuring that each side's length is accounted for in the total perimeter calculation.

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