How to Find the Perimeter of Right-Angled Triangle?

The perimeter of a geometrical shape is the total length of its boundary. In this step-by-step guide, you learn how to find the perimeter of a right-angled triangle.

How to Find the Perimeter of Right-Angled Triangle?

The perimeter of a right triangle is the sum of the lengths of all three sides, including the hypotenuse, height, and base.

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Step by step guide to finding the perimeter of the right-angled triangle

The perimeter of a right triangle is the sum of its sides. For example, if \(a, b\), and \(c\) are sides of a right-angled triangle, its perimeter would be: \((a + b + c)\). Now that the triangle is right-angled, we can say that its perimeter is the sum of the lengths of the two sides and the hypotenuse.

See the figure below, which shows a right-angled triangle with sides \(a, b\), and \(c\), where \(c\) is the hypotenuse and \(a, b\) are the sides that together form a \(90°\) angle. The three sides can be named as follows: \(a =\) height, \(b =\) base, and \(c =\) hypotenuse.

How to find the perimeter of a right triangle?

There are several ways to find the perimeter of a right triangle. We must check the parameters according to the given conditions to do this. Let’s see the different methods.

Method 1:

Knowing the length of all sides of a right triangle is enough to add their length. For example, if \(p, q,\) and \(r\) are the given sides, then:

Perimeter \(\color{blue}{= p + q + r}\)

This method is only possible if the measurement of all sides is known.

Method 2:

If the lengths of the sides are not given but the right triangle is drawn to scale, we use a ruler to measure the sides and add the dimensions of each side. The perimeter of the right-angled triangle is:

Perimeter \(\color{blue}{= sum\:of\:all\:the\:sides\:measured\:by\:the\:ruler}\)

Method 3: 

When both sides of a right triangle are given, we first find the missing side using the Pythagorean theorem and then calculate the perimeter of the right triangle. Pythagoras’s theorem states that the square of the hypotenuse length equals the sum of the squares of the other two sides of the right triangle.

\(\left(Hypotenuse\right)^2\:=\:\left(Base\right)^2\:+\:\left(Height\right)^2\)

See the triangle below, where \(a\) and \(b\) are sides that make a \(90°\) angle together, and \(c\) is the hypotenuse. For this purpose, the Pythagorean theorem is written as follows:

\(c^2\:=\:a^2\:+\:b^2\)

The following formulas can be derived to find the missing side:

  • \(a=\sqrt{c^2-b^2}\)
  • \(b=\sqrt{c^2-a^2}\)
  • \(c=\sqrt{a^2+b^2}\)

After finding the missing side, the perimeter of the right triangle can be calculated with the same basic formula: \(P = a + b + c\)

When one of the sides is not known, the following formulas can be used directly:

  • Perimeter \(\color{blue}{=\:a\:+\:b\:+\sqrt{a^2+b^2}\:}\), (when \(c\) is not given)
  • Perimeter \(\color{blue}{=\:b\:+\:c\:+\sqrt{c^2-b^2}\:}\), (when \(a\) is not given)
  • Perimeter \(\color{blue}{=\:c\:+\:a\:+\sqrt{c^2-a^2}\:}\), (when \(b\) is not given)

Finding the Perimeter of the Right-Angled Triangle – Example 1:

If the base is \(5\) units and the hypotenuse is \(13\) units, find the perimeter of a right triangle.

Solution:

Use the Pythagorean theorem to find the height:

\(\left(Hypotenuse\right)^2\:=\:\left(Base\right)^2\:+\:\left(Height\right)^2\)

\(\left(13\right)^2=\left(5\right)^2+\left(a\right)^2\)

\(\left(13\right)^2-\left(5\right)^2=\left(a\right)^2\)

\(169-25=144\)

\(a=\sqrt{144}=12\)

Height \(= 12\) units

The perimeter of the right triangle \(= 5 + 13 + 12 = 30\) units.

Exercises for Finding the Perimeter of the Right-Angled Triangle

Find the perimeter of each Right-Angled Triangle.

This image has an empty alt attribute; its file name is answers.png
  1. \(\color{blue}{43\space cm}\)
  2. \(\color{blue}{24\space cm}\)

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