Perimeter of a Triangle – Formula – Solved Examples

Perimeter of a triangle

The perimeter of a triangle is equal to the total sum of the measure of all three sides of that triangle. 

The formula for perimeter is different in different types of triangles based on their properties. We are going to derive the perimeter formula for the following triangles.

1. Equilateral Triangle
2. Isosceles Triangle
3. Scalene Triangle
4. Right-angled Triangle
5. Isosceles right-angled Triangle

Perimeter of an equilateral triangle

In an equilateral triangle, the measure of all sides is equal to each other.

In △PQR

PQ = QR = RP = s = measure of each side

Perimeter = PQ + QR + RP

                   = s + s + s

                   = 3s

Hence the perimeter of an equilateral triangle is equal to 3s where “s” is the length of each side.

Perimeter of an isosceles triangle

In an isosceles triangle, the measure of the two sides is equal to each other.

In △PQR

PQ = RP = s 

QR = a

Perimeter = PQ + QR + RP

                   = s + s + a

                   = 2s + a

Hence perimeter of an isosceles triangle is equal to (2s + a) where “s” is the measure of one of the two equal sides and “a” is the measure of the third side.

Perimeter of a scalene triangle

In a scalene triangle, the measure of all sides is unequal.

In △PQR

PQ = r

QR = p

RP = q

Perimeter = PQ + QR + RP

                   = r + p + q

                   = p + q + r

Hence the perimeter of a scalene triangle is equal to (p + q + r), where “p”, “q”, and “r” are the measure of the three sides.

Perimeter of a right-angle triangle

 In △PQR

∠PQR = 90°

PQ = r = leg

QR = p = leg

RP = hypotenuse = h

h=\sqrt{p^{2}+r^{2}} (pythagoras theorem)

Perimeter = PQ + QR + RP

                   = r + p + h

                   = r+p+\sqrt{p^{2}+r^{2}}

Hence the perimeter of a right-angle triangle is equal to \left(r+p+\sqrt{p^{2}+r^{2}}\right) where “p” and “q” are the measure of the legs of the right-angle triangle.

Perimeter of an isosceles right triangle

In an isosceles right triangle, the two sides other than hypotenuse are equal to each other

In △PQR

∠PQR = 90°

PQ = s = leg

QR = s = leg

RP = hypotenuse = h

h=\sqrt{s^{2}+s^{2}}

Perimeter = PQ + QR + RP

                   = s + s + h

                   = s+s+\sqrt{s^{2}+s^{2}}

                   = 2 \mathrm{~s}+\sqrt{2} \mathrm{~s}

                   = s(2+\sqrt{2})

Hence the area of an isosceles right-angle triangle is equal to \left[s(2+\sqrt{2})\right], where s is the measure of equal legs.

Solved Examples

1. In a right-angled triangle, two sides are equal, having a length of 7 cm. Find the perimeter of this triangle?

Solution:

Since two sides of the right-angled triangle are equal, it is an isosceles right triangle 

Length of the leg = s = 7 cm

Hence the perimeter of the triangle is equal to (14+7 \sqrt{2}) \mathrm{cm}.

2. The perimeter of △PQR is equal to 27 inches. Find the measure of each side of the triangle if it is an equilateral triangle?

Solution:

Since it is an equilateral triangle, the length of all the sides are equal to each other

 s = length of each side

Perimeter  = 3s

⇒ 3s = 27 inches

⇒ s = 27/3 inches = 9 inches

Hence the length of each side is equal to 9 inches.