Right Angle Formula: Area, Perimeter & Trigonometric Values

Right Angle formula can be defined as an angle that is equal to 90° or π/2 radian. A Right-angled Triangle is shaped as a polygon and contains three vertices and sides. The other kinds of triangles are isosceles, equilateral, and scalene triangles. A triangle can be defined as a simple closed geometrical figure that has been enclosed by three line segments. It is simply a three-sided polygon, with three vertices, and three angles.

• A triangle with one angle of 90 degrees is known as a right-angled triangle.
• The three sides of a triangle are known as Base (Adjacent), Perpendicular and Hypotenuse (Opposite).
• The side that makes a right angle with the triangle’s base is known as the perpendicular.
• The largest side is the hypotenuse which is found in the opposite direction of the right-angle.
• The adjacent side to the right angle is known as the legs (cathetus). So, both the Perpendicular and the base are the Cathetus.

Key Terms: Right-Angle, Right-Angle Triangle, Angle, Hypotenuse, Base, Altitude, Obtuse Angles, Acute Angles, Isosceles Triangle, Scalene Triangle, Equilateral Triangle

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What is Right Angle?

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The right angle is equal to 90 degrees. For a right-angled triangle, the three sides it has are known as: Perpendicular, Base(Adjacent) and Hypotenuse(Opposite). Perpendicular is the side that makes right angle with the base of the triangle.

• The largest side opposite to the right angle (90°) is also referred to as the Hypotenuse.
• The adjacent side to the right angle are also known as the legs (cathetus). Which is why both the base and perpendicular are called Cathetus.
• A right angle’s measure is written in degrees like 90° or in radians as π/2 (1.5708) radians.

In order to determine the side of the triangle, it is important to note the other sides of the triangle. The figure below demonstrates the same:

Right-Angled Triangle Diagram

The Right angled triangle formula is also known by the name of Pythagorean theorem (Pythagoras Theorem) and can be represented by:

The formula can also be represented as: “The square of the hypotenuse is equivalent to the sum of the squares of the other two sides”. 

Trigonometric Functions

Also Read: 

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Right-Angle Triangle Definition

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A right-angled triangle is a triangle that has an angle between the perpendicular and the base which is equal to 90 degrees. It contains three sides, which are known as the “base”, the “perpendicular” and the “hypotenuse”. A right-angled triangle is trigonometry’s foundation and also one of the basic shapes in geometry.

Right-angled triangle

As the given image shows, triangle ABC is a right triangle, which has the base, altitude, and hypotenuse. In the given image, AB is the base, BC is the altitude, and AC is the hypotenuse. The hypotenuse, in a right triangle, is known to be the largest side and is opposite to the right angle.

Also Check-Out: 2 Cos A Cos B Formula

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Right-Angled Triangle Formula

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To determine whether a triangle is a right triangle or not, we use a formula derived from the Pythagoras theorem. According to the theorem, the sum of the squares of the base and the height of a triangle is equal to the square of the hypotenuse.

It can also be written as:

→ (Hypotenuse)² = (Base)² + (Perpendicular)²

That is, H² = B² + P²

Where,

• H = Hypotenuse, meaning the longer side of a right-angled triangle.
• B = Base, meaning the side where the right-angle triangle is erected upon.
• P = Perpendicular, meaning the straight-line that forms the right angle.

Read Also: Cotangent formula

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Right-Angled Triangle Properties

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There are several properties of right-angled triangle, 

• In a right-angled triangle, one of the angles’ measures exactly equal to 90 degrees.
• The other angles present in the right triangle other than the right angle must be less than 90 degrees (a.k.a must be acute angles).
• The other two sides which are adjacent to the right angle are called base and perpendicular.
• The right Triangle’s circumcentre passes through all three vertices of the triangle.
• The radius of such a circle is equal to half of the hypotenuse’s length.
• If one angle in the triangle is 90° and the other two angles are 450 each, then such a triangle will be known as an Isosceles Right Angled Triangle. In which the adjacent sides to angles with 90° are equal in length.
• The perimeter of a right triangle is calculated by adding the length of the three sides. S = a + b + c. Where the sides are a,b and c respectively.

These were some significant right triangle properties.

Also Read: Secant Square x Formula

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Area of Right-Angled Triangle

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The area of a right triangle is the area inside the three sides of the triangle in a fixed plane. The formula that is used to find out a right-angled triangle’s area is given below:

Here, The Height is the same as the Perpendicular.

Area Formula of Right-Angle Triangle

Check More: Area of a Triangle

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Perimeter of Right Angle Triangle

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The right triangle perimeter can simply be defined as the sum of the measures of all three sides. It is the total sum of the base, altitude, and hypotenuse of a right triangle. Thus, 

The formula to determine the perimeter of right triangle is:

Here, a, b and c are the measure of its three sides.

The perimeter of a right triangle is equal to the sum of the sides, BC + AC + AB = (a + b + c) units. The perimeter consists of a linear value and has a unit of length.

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Types of Right Triangles

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There are two Right Triangle Types, namely:

Right-Angled Isosceles Triangle

For a Right-Angled Isosceles Triangle:

• When two sides of a right-angled triangle, other than the hypotenuse, like the perpendicular and the base are congruent, then it is known as a right-angled isosceles triangle or an isosceles right triangle.
• In such a triangle, the angles produced by the perpendicular and the base with the hypotenuse are congruent.
• So the measure of both of these angles is 45 degrees.

Scalene Right Triangle

For a Scalene Right Triangle:

• A scalene right triangle contains one angle which is 90°, alongside the other two angles which are of different measurements.
• A Scalene triangle also has a special case, that is, 30º-60º-90º, which also is a right triangle that has the ratio 2:1 of the longest side of the triangle to its shortest side.
• The side opposite to the 30º angle is known to be the shortest side.

Also Read: Multiple Angle Formulas

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Trigonometric Values of Right Triangle

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The values of the various trigonometric functions at 90 degrees of angle, in trigonometry, is shown by:

• Cos 90° = 0
• Sin 90° = 1
• Cot 90° = 0
• Tan 90° = Not defined
• Cosec 90° = 1
• Sec 90° = Not defined

Also Read: Unit Circle Formula

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Things To Remember

• A Right Angle is an angle which is known to be equal to 90 degrees or π/2 radian. 
• A right-angled Triangle is shaped as a polygon and it has the minimum number of sides among the many shapes. 
• The other forms of triangles are isosceles, equilateral, right-angled and scalene triangles.
• To determine the state of a given triangle, whether it is a right triangle or not, we use a formula derived from the Pythagoras theorem. 
• The formula of Pythagoras Theorem is = (Hypotenuse)² = (Adjacent Side)² + (Opposite side)²

Read More: 

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