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Right Angled Triangle: Properties & Formulas

TL;DR

A right angled triangle is a triangle with one angle of exactly 90°, and the side opposite that angle, the hypotenuse, is always the longest. This article covers its definition, properties, the Pythagorean theorem, area and perimeter formulas, the special 45-45-90 and 30-60-90 types, six worked examples, and the mistakes students make most.

The One Angle That Builds Bridges

Every staircase, roof truss, and ramp you have ever used was checked against a single shape: the triangle with one square corner. That square corner is not a coincidence of design, it is the property that lets builders turn a measured horizontal and a measured vertical into an exact slanted length, every time. Once you see why that one angle fixes everything else, the formulas stop being a list to memorise and become a single connected idea.

What Is a Right Angled Triangle?

A right angled triangle (also called a right triangle) is a triangle in which one interior angle measures exactly 90°. The two sides that form the right angle are the legs (sometimes called base and height), and the side opposite the right angle is the hypotenuse, the longest of the three sides. Because the three angles of any triangle add to 180°, the two angles that are not the right angle must add to 90°. So both of them are acute, and a right angled triangle can never hold a second right angle or any obtuse angle. The right angle is always the largest angle in the figure.

Properties of a Right Angled Triangle

The defining 90° angle forces a short list of properties that hold for every right triangle, with no exceptions:

• One right angle, two acute angles. The acute pair always sums to 90°, so they are complementary.
• The hypotenuse is the longest side. It sits opposite the largest angle (the right angle), and it is always longer than either leg.
• The Pythagorean relationship holds. The squares of the two legs add to the square of the hypotenuse, every time.
• The circumcentre sits at the midpoint of the hypotenuse. A right triangle's circumscribed circle has the hypotenuse as its diameter, so the midpoint of the hypotenuse is equidistant from all three vertices.
• The altitude to the hypotenuse is the geometric mean of the two segments it creates. If that altitude splits the hypotenuse into pieces of length p and q, then its height is h = \sqrt{pq}.

The Pythagorean Theorem

The most useful property earns its own heading. In a right angled triangle, the Pythagorean theorem states that the square of the hypotenuse equals the sum of the squares of the two legs:

c² = a² + b², where a and b are the legs and c is the hypotenuse. To find the hypotenuse, take the square root of both sides: c = \sqrt{a² + b²}. To find a missing leg, rearrange: a = \sqrt{c² - b²}.

Area and Perimeter of a Right Angled Triangle

The right angle does something convenient for area: the two legs are already perpendicular, so they serve directly as base and height.
Area. The area of a right angled triangle is half the product of its two legs:

Area = \frac{1}{2} \times a \times b, where a and b are the legs (the base and height).
For example, legs of 6 cm and 8 cm give an area of \frac{1}{2}(6)(8) = 24 cm².

Perimeter. The perimeter is simply the sum of all three sides:

Perimeter = a + b + c, where c is the hypotenuse. If only the two legs are known, find c with the Pythagorean theorem first, then add.

What Are the Types of Right Angled Triangles?

A common reader question is whether all right triangles are the same shape, and they are not. Based on the two acute angles, right triangles split into two families, with one famous special case:

• Isosceles right triangle (45-45-90). The two legs are equal, so the two acute angles are both 45°.
• Scalene right triangle. All three sides have different lengths and the two acute angles are different.
• The 30-60-90 triangle. A special scalene right triangle whose sides are always in the ratio 1:3:2.

Examples of Right Angled Triangle

Example 1: Find the area of a right angled triangle with legs 9 cm and 12 cm

Area = \frac{1}{2}(9)(12) = 54 cm².

Example 2: A right angled triangle has legs of 5 cm and 12 cm. Find the hypotenuse

c = \sqrt{5² + 12²} = 13 cm.

Example 3: Check whether sides of 11, 60, and 61 inches form a right angled triangle

11² + 60² = 61² → true.

Example 4: A right angled triangle has one leg 8 cm and hypotenuse 17 cm. Find the other leg and the perimeter

Using Pythagorean theorem, find the other leg = 15 cm; perimeter = 40 cm.

Example 5: An isosceles right triangle has legs of 7 cm each. Find its hypotenuse and area

Hypotenuse = 7√2 ≈ 9.9 cm; Area = 24.5 cm².

Example 6: The altitude from the right angle meets the hypotenuse and splits it into segments of 4 cm and 9 cm. Find the length of that altitude

h = \sqrt{4 \times 9} = 6 cm.

Why the Right Angled Triangle Matters

A shape earns its place in every syllabus by what it unlocks, and this one unlocks more than almost any other.

Where Students Trip Up on Right Angled Triangles

Mistake 1: Adding the legs to get the hypotenuse

Don’t do this: Add the legs, instead of using Pythagorean theorem. The correct way: Square the legs, add, then take the square root: c = \sqrt{a² + b²}.

Mistake 2: Treating the hypotenuse as a leg in the area formula

Don’t do this: Use hypotenuse in area calculation. The correct way: Use the two legs as base and height: Area = \frac{1}{2} \times a \times b.

Mistake 3: Picking the wrong side as the hypotenuse

Don’t do this: Assume the longest number is the hypotenuse. The correct way: The hypotenuse is always opposite the right angle and is the longest.

Key Takeaways

• A right angled triangle has one 90° angle and two complementary acute angles.
• The Pythagorean theorem relates the legs to the hypotenuse.
• Area is \frac{1}{2} \times a \times b; perimeter is the sum of all sides.
• The two main types are the isosceles right triangle (45-45-90) and the scalene right triangle.