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What is Right Triangle?

A triangle in which one and only one of the interior angles is 90° is called a right triangle. The longest side of the right triangle, which is also the side opposite the right angle, is the hypotenuse and the two arms of the right angle are the height and the base.

In a right triangle, the side that is opposite of the 90° angle is the longest side of the triangle, and is called the hypotenuse.

The sides of a right triangle are commonly referred to with the variables a, b, and c, where c is the hypotenuse and a and b are the lengths of the shorter sides.

Their angles are also typically referred to using the capitalized letter corresponding to the side length: angle A for side a, angle B for side b, and angle C (for a right triangle this will be 90°) for side c, as shown below.

Here’s what a right triangle looks like:

What is The Formula for a Right-Angled Triangle?

Pythagorean Theorem

Pythagoras discovered that the hypotenuse square equals the sum of the squares of the other two sides in a right-angled triangle.

c² = a² + b²

How to calculate the Area of a right triangle?

The formula for the area is: Area = (1/2) a × b = (1/2) c × h_c

Where a is Base, b is Perpendicular, c is Hypotenuse, h_c is the Altitude of side c

How to calculate the perimeter of a right triangle?

The formula for the perimeter is: Perimeter of right triangle = Length of (Base + Perpendicular + Hypotenuse)

P (perimeter) = a + b + c (sum of the sides of a triangle)

How to calculate the Altitude of side c (h_c)| Given a, b, c in a right-angled triangle?

Given side lengths a, b, c calculate Altitude of side c (h_c) in a right-angled triangle

h_c = a × b / c

How to calculate the Inradius(Inscribed Circle Radius) of a right triangle?

Inradius = a × b / ( a + b + c )

How to calculate the Circumradius(Circumscribed Circle Radius) of a right triangle?

Circumradius = c / 2

How to calculate the Angle bisector of a right triangle?

Angle bisector of A

2 × b × c × Cos(A/2) / (b + c)

Angle bisector of B

2 × a × c × Cos(B/2) / (a + c)

Angle bisector of C (t_c)

t_c² = 2 × a² × b² / (a + b)²

How to calculate the Median of a right triangle?

Median of side a ( m_a )

m_a² = b² + 0.25 × a²

Median of side b ( m_b )

m_b² = a² + 0.25 × b²

Median of side c( m_c )

m_c = c / 2

How to calculate the angle of a right triangle?

sin(A) = a / c so A = arcsin(a / c) (inverse sine)

sin(B) = b / c so B = arcsin(b / c) (inverse sine)

How to calculate the circumcenter of a right angle triangle?

For a right-angled triangle, the circumcenter, i.e., the center of the circle circumscribed on the triangle, coincides with the midpoint of the triangle's longest side (its hypotenuse).

How to calculate the orthocenter of a right angle triangle?

The orthocenter of a right-angled triangle, i.e., the point where the triangle's altitudes intersect, coincides with the triangle's vertex of the right angle.

What is the Types of Right Triangles?

1. Isosceles right triangle

2. Scalene right triangle

What is the Properties of a Right Triangle?

1. The right angle is always the largest angle in a right triangle.

2. The hypotenuse, the side opposite the right angle, is the longest side.

3. There can’t be any obtuse angles in a right triangle.

4. If a right triangle is isosceles (i.e., its two non-hypotenuse sides are the same length) it has one line of symmetry. Otherwise, the triangle will have no lines of symmetry.

5. A right triangle cannot have all 3 sides equal

6. Not all right angled triangles are similar