Perimeter of right-angled triangle

The perimeter of a triangle can be defined as the sum of all three sides. In most cases, the formula for perimeter is straightforward as a+b+c. But, in some cases, one or two sides might be unknown. Thus, we have to apply some theorems or use some formulas to find the perimeter. In this article, we look at methods on how to find out the perimeter of a right-angled triangle.

The formula for the perimeter of right angle triangle

1. Simple method: Add all sides
2. Theorem: Use the Pythagoras theorem to calculate the hypotenuse.
3. Use law of cosines to find unknown side
4. Use the law of sine [a sin A = b sin B = c sin C]
5. Use the formula for the area

1) Simple method: Add all sides

This is the basic method to find the perimeter of a right-angle triangle. Add all sides of the triangle.

Formula for perimeter of right angled triangle = a + b+ c

For example in the below case:

a = 4cm; b = 3 cm and c = 5 cm

The perimeter of this right angle triangle will be = a + b + c = 4 + 3 + 5 = 12 cm. The unit of perimeter will be always like cm, m, km etc.

2) Using Pythagoras theorem

Sometimes it happens that we do not know all the sides of a right-angled triangle. Suppose if we do not know the side opposite to the right angle (90^o), then we can use the Pythagoras theorem to find it.

Pythagoras theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

c² = a² + b²

In the above image, a = 4; b= 3; c is unknown. We can find this by the Pythagoras theorem.

c² = 4² + 3²

c = sqrt(25)

c = 5

3) Using law of cosines

This law can be used when anyone’s side is unknown and we know the measure of the opposite angle. This law can be applied to any triangle in general.

In the above equation, the unknown side is c, a and b are known, and y is the angle opposite to the unknown side. If c is the hypotenuse, then y will be 90^o. As cos(90) is zero, hence the equation reduces to sqrt(a² + b²). This proves that this law is consistent with Pythagoras theorem.

For example in the above image, two sides are known as 3 cm and 4.24 cm (hypotenuse), the side with length x is unknown. Applying the law of cosine we get:

x = sqrt {(3)² + (4.24)² – 2*3*4.24 } = ~3 cm

Let us verify our answer using pythagores theorem:

LHS : square of hypotenuse = (4.24)² = ~18

RHS : sum of square of other 2 sides = (3)² + (3)² = 18

LHS = RHS. Hence our answer is verified.

Now, we can calculate the perimeter of the right-angled triangle, which is the sum of three sides: 3 + 3 + 4.24 = 13.24 cm.

You can use this tool to calculate the cosine values.

4) Using law of sines

This law can be used when at least two angles are known along with two sides. This law can also be used in combination with other laws and formulas to find the perimeter of any triangle. In a right-angled triangle, we already know one angle as 90^o, so this law can be preferred for calculating the perimeter of right-angled triangles.

In the above example, two sides and two angles are known. Let us calculate the unknown side x by using the law of sines.

sin(45)/x = sin(45)/3

x = 3

Let us verify our answer by using the other side also.

sin(45)/x = sin(90)/4.24

0.707/x = 1/4.24

x = 0.707 *4.240 = 3 cm

Hence our answer is verified. This method is very useful when two angles and at least one side are known. The perimeter of this right-angled triangle can now be calculated as 3 + 3 + 4.24 = 10.24 cm.

You can use this tool to calculate sine values.

5) Using the area formula for a right-angled triangle

We saw cases where all three sides are known or at least two sides are known. There will be cases when only one side is known. Then it requires us to use 2 or more formulas to arrive at the perimeter of a right angled triangle.

We know that the formula for the area of a right-angled triangle is ab/2. This equation can be used with other laws when two sides are known. We can then solve two equations with two variables to get the other two sides and then accordingly calculate the perimeter of the right-angled triangle.

In the above example, area = 4 cm² and one side is known as 3 cm.

Formula for area of a right angled triangle = a*b/2

6 = 1/2 * 3 * (x)

x = 4 cm

Now let us calculate the hypotenuse using Pythagoras theorem:

H² = 3² + 4²

H = 5 cm

Hence the perimeter of this triangle will be: 3 + 4 + 5 = 12 cm.

Numerical problems on the perimeter of a right angled-triangle

Find the perimeter of the right-angled triangles shown below:

Here we know two angles(A =60 and B = 30) and one side(c=12). We can use law of sines in this scenario:

a/sin(60) = 12/sin(90)

a/0.866 = 12/1

a = 10.4 cm

Here the area of the right angled triangle is known and also the length of one side is known (b=12).

Area= ab/2 = (a*12)/2

36 = a*6

a = 6 cm

using Pythagoras theorem,

c = sqrt(144 + 36)

c = 13.4 cm

Perimeter = a + b + c = 12 + 6 + 13.4 = 31.4 cm

Here we are provided two sides (c = 7.01 cm and b = 5 cm) and one angle (A =45^o). Here we can apply the law cosine to find the unknown side.

a = sqrt(49.14 + 25 – 24.5)

a = 5.9

Hence the perimeter of the right angled triangle will be = 5 + 5.9 + 7.01 = 16.91 cm

Here we are one angle (B = 45^o) and one side (c= 7.01 cm). We can use the law of sines to find the other two sides. We can find the other angle A = 45^o (by using A +B + C = 180).

Sin45/a = sin(90)/7.01

a = ~5 cm

sin45/b = sin(90)/7.01

b = ~5 cm

Hence, perimeter = 5 + 5 + 7.01 = 17.01 cm

See Also

Area of a sector formula
Perimeter of a sector-Formula
Every square is a rectangle
Orthocenter of a triangle
Volume of a parallelepiped