What is a Right Triangle?
A triangle with a right angle is known as a right triangle. The sides of the right triangle forming the arms of the right angle are known as the base and perpendicular. The base and perpendicular of the triangle are also called its legs. The longest side of a right triangle opposite the right angle is called the hypotenuse of the right triangle.
[Note: The perpendicular of a right triangle is also referred to as its height.]
Let us explore the formulas related to the right triangle in the upcoming sections of this article.
Formula for the Hypotenuse of a Right Triangle
The Pythagorean theorem describes the relationship between the sides of a right triangle. It states that: ‘In a right triangle the sum of the squares of the lengths of the base and perpendicular is equal to the square of the length of the hypotenuse’. Mathematically, this can be written in the form of an equation:
\( Hypotenuse^2~=~Base^2~+~Perpendicular^2\)
This relationship forms the basis for the formula for the hypotenuse of a right triangle, which is given by:
\( Hypotenuse~=~\sqrt{Base^2~+~Perpendicular^2}\)
Formula for the Perimeter of a Right Triangle
Like any other triangle, the perimeter of a right triangle is the sum of the lengths of all its three sides.
Perimeter, \( P~=~(b~+~p~+~H)\)
Where b, p and H are the measures of three sides of the right triangle which are, the base, the perpendicular and the hypotenuse.
Perimeter is measured in the same units as the measure of length.
Formula for the Area of a Right Triangle
Area is the measure of the space enclosed by the boundary of any closed shape. The formula for the area of a right triangle is:
Area, \( A~=~\frac{1}{2}~bh\)
Where, b = base of the triangle
h = height of the triangle
So, the area of a right triangle is half of the product of its base and height, or perpendicular.
Area, for any closed shape, is always measured in square units.
Solved Examples
Example 1: Find the area of a right triangle if its base is 8 cm and height is 5 cm.
Solution:
Area, \( A~=~\frac{1}{2}~bh\) [Formula for the area of a right triangle]
\( =~\frac{1}{2}~\times~8~\times~5\) [Substitute 8 for b and 5 for h]
= 4 × 5 [Simplify]
= 20 [Simplify further]
Therefore the area of the given right triangle is 20 square centimeters.
Example 2: The perimeter of a right-angled triangle is 35 cm. Its height and hypotenuse measure 10 cm and 16 cm respectively. Find its area.
Solution:
Here we have been given the perimeter, height and hypotenuse of the triangle.
To determine the area we first need to find the base of the triangle. Let us use the formula for the perimeter of the right triangle to find the base:
Perimeter, \( P~=~(b~+~p~+~H)\)
\( 35~=~(b~+~10~+~16)\) [Substitute the given values]
\( b~=~35~-26\) [Solve for b]
\( b~=~9\) [Subtract]
Area, \( A~=~\frac{1}{2}~bh\) [Formula for the area of a right triangle]
= \( \frac{1}{2}~\times~9~\times~10\) [Substitute 9 for b and 10 for h]
= 9 × 5 [Simplify]
= 45 [Simplify further]
So, the area of the right triangle is 45 square centimeters.
Example 3: Find the height of a right triangle with a base measuring 5 inches and the hypotenuse measuring 13 inches.
Solution:
\( Hypotenuse^2~=~Base^2~+~Perpendicular^2\) [Formula for the hypotenuse of a right triangle]
\( 13^2~=~5^2~+~Perpendicular^2\) [Substitute the given values]
\( 169~=~25~+~Perpendicular^2\) [Square of 13 and 5]
\( Perpendicular~=~\sqrt{169~-~25} \) [Solve for perpendicular]
\( Perpendicular~=~\sqrt{144} \)
\( =12 \) [Take positive square root of 144]
So, the perpendicular or the height of the right triangle is 12 inches.
A right triangle has exactly two acute angles.
A right triangle in which the base and perpendicular are equal in length is known as an isosceles right triangle.
The Pythagorean theorem can be used to:
- Determine whether a given triangle is right angled or not.
- Determine the length of a missing side of a right triangle.