Right Angled Triangle | Properties and Formula
Last Updated :
11 Aug, 2025
Right Angle Triangle is a type of triangle that has one angle measuring exactly 90 degrees or right angle (90°). It is also known as the right triangle. In a right triangle, the two shorter sides called the perpendicular and the base, meet at the right angle (90°), while the longest side, opposite the right angle, is called the hypotenuse. Right triangles are used in many areas, from construction to navigation, and play a key role in trigonometry.
In this article, we will learn about the right-angle triangle, its definition, perimeter, area, right-angled triangle formula, and its properties in detail.
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Right Angled TriangleWhat is a Right Angled Triangle?
A right-angled triangle is a type of triangle that has one of its angles measuring exactly 90 degrees. This 90-degree angle, also known as a right angle, gives the right-angled triangle its name and distinct properties.
A triangle with any interior angle equal to 90° is called a Right Triangle.
Sum of all the interior angles of the triangle is 180° which is called the Angle Sum Property of a Triangle. So if any one triangle is 90° the sum of the other two angles is also, 90°.
Properties of Right Angled Triangle
A Right Angled Triangle has the following key properties :
- One of the angles in a right-angled triangle is exactly 90 degrees.
- Side opposite the right angle is the longest side of the triangle and is called the hypotenuse.
- For triangles with the same angles, the sides are in a consistent ratio. For example, in a 45-45-90 right triangle, the sides are in the ratio 1:1:√2, and in a 30-60-90 triangle, the sides are in the ratio 1:√3/2.
- Altitude drawn to the hypotenuse of a right triangle creates two smaller right-angled triangles, each of which is similar to the original right-angled triangle.
- Every right-angled triangle has a circumcircle (circle passing through all three vertices) with the hypotenuse as its diameter. It also has an incircle (circle tangent to all three sides), with the center at the intersection of the angle bisectors.
Formula for right-angled triangle is given by the Pythagoras Theorem. According to the pythagoras theorem, in a right-angled triangle, the square of the hypotenuse is equal to sum of the squares of the other two sides.
(Hypotenuse)2 = (Perpendicular)2 + (Base)2
Perimeter of Right Angled Triangle
The perimeter of the right triangle shown above is equal to the sum of the sides,
BC + AC + AB = (a + b + c) units.
Perimeter of Triangle = (a + b + c) units
The perimeter is a linear value with a unit of length. Therefore,
Area of Right Angled Triangle
Area of a right triangle is the space occupied by the boundaries of the triangle.
Area of a right angle triangle is given below,
Area of a Right Triangle = (1/2 × base × height) square units.
Also View:
For any right angle triangle, PQR right angled at Q with hypotenuse as, PR
Now if we flip the triangle over its hypotenuse a rectangle is formed which is named PQRS. The image given below shows the rectangle form by flipping the right triangle.
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As we know, the area of a rectangle is given as the product of its length and width, i.e. Area = length × breadth
Thus, the area of Rectangle PORS = b x h
Now, the area of the right angle triangle is twice the area of the rectangle then,
Thus,
Area of ∆PQR = 1/2 × Area of Rectangle PQRS
A = 1/2 × b × h
Hypotenuse of Right Angled Triangle
For a right triangle, the hypotenuse is calculated using the Pythagoras Theorem,
H = √(P2 + B2)
where,
- H is Hypotenuse of Right Triangle
- P is Perpendicular of Right Triangle
Solved Examples Questions
Let's solve some example problems on right angled triangles.
Example 1: Find the area of a triangle if the height and hypotenuse of a right-angled triangle are 10 cm and 11 cm, respectively.
Solution:
Given:
- Height = 10 cm
- Hypotenuse = 11 cm
Using Pythagoras' theorem,
(Hypotenuse)2 = (Base)2 + (Perpendicular)2
(11)2 = (Base)2 + (10)2
(Base)2 = (11)2 - (10)2 = 121 - 100
Base = √21 = 4.6 cm
Area of the Triangle = (1/2) × b × h
Area = (1/2) × 4.6 × 10
Area = 23 cm2
Example 2: Find out the area of a right-angled triangle whose perimeter is 30 units, height is 8 units, and hypotenuse is 12 units.
Solution:
- Perimeter = 30 units
- Hypotenuse = 12 units
- Height = 8 units
Perimeter = base + hypotenuse + height
30 units = 12 + 8 + base
Base = 30 - 20 = 10 units
Area of Triangle = 1/2×b×h = 1/2 ×10 × 8 = 40 sq units
Example 3: If two sides of a triangle are given find out the third side i.e. if Base = 3 cm and Perpendicular = 4 cm find out the hypotenuse.
Solution:
Given:
- Base (b) = 3 cm
- Perpendicular (p) = 4 cm
- Hypotenuse (h) = ?
Using Pythagoras theorem,
(Hypotenuse)2 = (Perpendicular)2 + (Base)2
= 42 + 32 = 16 + 9 = 25 cm2
Hypotenuse = √(25)
Hypotenuse = 5 cm
Important Maths Related Links:
Conclusion
Right triangle is an important shape in geometry, defined by one 90-degree angle. It is widely used in various real-world applications, from building structures to solving distance related problems. The relationship between the sides and angles of a right triangle , especially through the Pythagorean theorem, is essential for calculating lengths and understanding trigonometry.
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