Perimeter of Triangle
Imagine you want to put a fence around a triangle-shaped garden. To know how much fencing you need, you have to find the total length of the boundary. The total length of the boundary of a shape is called the perimeter.
The word "perimeter" comes from the Greek words "peri" (around) and "metron" (measure). So perimeter means "measure around" — exactly what we do when we add up all the sides!
For a triangle, the perimeter is simply the sum of all three sides. It is one of the easiest formulas in geometry. Whether the triangle is small or big, equilateral or scalene, the method is always the same — add the three sides.
In Class 6 Mathematics (NCERT), the perimeter of a triangle is studied in the chapter Mensuration. You will learn how to find the perimeter of different types of triangles — scalene, isosceles, and equilateral — and solve real-life problems about fencing, framing, and wires.
What is Perimeter of Triangle?
Definition: The perimeter of a triangle is the total length of its three sides added together.
Key terms:
- Perimeter: The total distance around a shape. For a triangle, it is the sum of its three sides.
- Side: Each of the three straight edges of a triangle.
- Unit: Perimeter is measured in units of length — cm, m, km, inches, etc.
Types of triangles and their perimeters:
- Scalene triangle: All three sides are different. Perimeter = a + b + c.
- Isosceles triangle: Two sides are equal. If the equal sides are a and the third side is b, Perimeter = 2a + b.
- Equilateral triangle: All three sides are equal. If each side is a, Perimeter = 3a.
Perimeter of Triangle Formula
Perimeter of a Triangle:
Perimeter = a + b + c
Where a, b, and c are the lengths of the three sides.
For an equilateral triangle (all sides equal = a):
Perimeter = 3a
For an isosceles triangle (two equal sides = a, third side = b):
Perimeter = 2a + b
To find a missing side:
- If perimeter and two sides are known: third side = Perimeter − (sum of two known sides).
- Missing side = P − a − b
Derivation and Proof
Understanding the perimeter formula:
Why perimeter = sum of all sides:
- Imagine an ant walking along the boundary of a triangle, starting from one corner.
- The ant walks along the first side (length a), then turns and walks along the second side (length b), then turns again and walks along the third side (length c) back to the start.
- The total distance the ant walked = a + b + c.
- That total distance is the perimeter.
Why equilateral perimeter = 3a:
- In an equilateral triangle, all three sides have the same length: a.
- Perimeter = a + a + a = 3 × a = 3a.
Why isosceles perimeter = 2a + b:
- An isosceles triangle has two equal sides (each of length a) and one different side (length b).
- Perimeter = a + a + b = 2a + b.
Think of it like a fence:
- If you need to fence a triangular park, you buy fencing equal to the perimeter.
- The amount of fencing = total length of all three sides.
Types and Properties
Types of perimeter problems:
1. Finding perimeter when all three sides are given:
- Add all three sides together.
- Example: sides 5, 7, 9 cm → Perimeter = 5 + 7 + 9 = 21 cm.
2. Finding perimeter of an equilateral triangle:
- Multiply one side by 3.
- Example: side = 8 cm → Perimeter = 3 × 8 = 24 cm.
3. Finding perimeter of an isosceles triangle:
- Double the equal side and add the base.
- Example: equal sides = 10 cm, base = 6 cm → Perimeter = 2(10) + 6 = 26 cm.
4. Finding a missing side:
- Subtract the known sides from the perimeter.
- Example: P = 30 cm, two sides are 12 cm and 9 cm → third side = 30 − 12 − 9 = 9 cm.
5. Finding the side of an equilateral triangle from perimeter:
- Divide the perimeter by 3.
- Example: P = 36 cm → side = 36/3 = 12 cm.
6. Cost-based problems:
- Find the perimeter first, then multiply by cost per unit length.
Solved Examples
Example 1: Example 1: Perimeter of a scalene triangle
Problem: Find the perimeter of a triangle with sides 8 cm, 11 cm, and 13 cm.
Solution:
Given:
- a = 8 cm, b = 11 cm, c = 13 cm
Perimeter:
- P = a + b + c
- P = 8 + 11 + 13
- P = 32 cm
Answer: The perimeter is 32 cm.
Example 2: Example 2: Perimeter of an equilateral triangle
Problem: Find the perimeter of an equilateral triangle with side 15 cm.
Solution:
Given:
- Each side = 15 cm (equilateral = all sides equal)
Perimeter:
- P = 3 × side
- P = 3 × 15
- P = 45 cm
Answer: The perimeter is 45 cm.
Example 3: Example 3: Perimeter of an isosceles triangle
Problem: An isosceles triangle has two equal sides of 12 cm each and a base of 8 cm. Find the perimeter.
Solution:
Given:
- Equal sides (a) = 12 cm
- Base (b) = 8 cm
Perimeter:
- P = 2a + b
- P = 2(12) + 8
- P = 24 + 8
- P = 32 cm
Answer: The perimeter is 32 cm.
Example 4: Example 4: Finding a missing side
Problem: The perimeter of a triangle is 42 cm. Two of its sides are 15 cm and 13 cm. Find the third side.
Solution:
Given:
- Perimeter = 42 cm
- Side 1 = 15 cm, Side 2 = 13 cm
Find the third side:
- Third side = Perimeter − Side 1 − Side 2
- = 42 − 15 − 13
- = 14 cm
Verify: 15 + 13 + 14 = 42 ✓
Answer: The third side is 14 cm.
Example 5: Example 5: Finding side of equilateral triangle from perimeter
Problem: The perimeter of an equilateral triangle is 27 cm. Find the length of each side.
Solution:
Given:
- Perimeter = 27 cm
- All three sides are equal (equilateral)
Find each side:
- Side = Perimeter ÷ 3
- Side = 27 ÷ 3
- Side = 9 cm
Verify: 9 + 9 + 9 = 27 ✓
Answer: Each side is 9 cm.
Example 6: Example 6: Cost of fencing
Problem: A triangular garden has sides 20 m, 25 m, and 30 m. Find the cost of fencing it at Rs 50 per metre.
Solution:
Step 1: Find the perimeter:
- P = 20 + 25 + 30 = 75 m
Step 2: Find the cost:
- Cost = Perimeter × Rate
- Cost = 75 × 50
- Cost = Rs 3,750
Answer: The cost of fencing is Rs 3,750.
Example 7: Example 7: Perimeter with decimal sides
Problem: Find the perimeter of a triangle with sides 5.5 cm, 7.3 cm, and 6.2 cm.
Solution:
Perimeter:
- P = 5.5 + 7.3 + 6.2
- P = 19 cm
Answer: The perimeter is 19 cm.
Example 8: Example 8: Finding the base of an isosceles triangle
Problem: An isosceles triangle has a perimeter of 50 cm and equal sides of 18 cm each. Find the base.
Solution:
Given:
- P = 50 cm, equal sides = 18 cm
Find the base:
- P = 2a + b
- 50 = 2(18) + b
- 50 = 36 + b
- b = 50 − 36 = 14 cm
Answer: The base is 14 cm.
Example 9: Example 9: Comparing perimeters
Problem: Triangle A has sides 10, 10, 10 cm. Triangle B has sides 8, 12, 10 cm. Which has a greater perimeter?
Solution:
Triangle A (equilateral):
- P = 10 + 10 + 10 = 30 cm
Triangle B (scalene):
- P = 8 + 12 + 10 = 30 cm
Answer: Both triangles have the same perimeter of 30 cm, even though they have different shapes.
Example 10: Example 10: Wire bent into an equilateral triangle
Problem: A wire of length 54 cm is bent into the shape of an equilateral triangle. What is the length of each side?
Solution:
Given:
- Length of wire = 54 cm
- The wire forms the perimeter of the equilateral triangle.
Find the side:
- Perimeter = 54 cm
- Side = 54 ÷ 3 = 18 cm
Verify: 18 + 18 + 18 = 54 cm ✓
Answer: Each side of the equilateral triangle is 18 cm.
Real-World Applications
Where is the perimeter of a triangle used?
- Fencing: Farmers fence triangular plots of land. The amount of fencing needed equals the perimeter.
- Framing: A triangular photo frame needs border material equal to its perimeter.
- Road around a park: If a park is triangular, the length of the walking path around it is the perimeter.
- Construction: Builders calculate the perimeter of triangular sections of roofs for installing gutters or trim.
- Art and craft: Making a triangular border for a bulletin board requires ribbon equal to the perimeter.
- Sports: Some running tracks or relay courses follow triangular paths. The distance of one lap equals the perimeter.
- Flag making: Triangular flags (like pennants) need stitching or binding along all three edges — that total is the perimeter.
Key Points to Remember
- The perimeter of a triangle = sum of all three sides = a + b + c.
- For an equilateral triangle (all sides equal): Perimeter = 3a.
- For an isosceles triangle (two equal sides): Perimeter = 2a + b.
- Perimeter is measured in units of length (cm, m, km), not square units.
- To find a missing side: subtract the known sides from the perimeter.
- To find the side of an equilateral triangle from perimeter: divide by 3.
- All sides must be in the same unit before adding.
- The perimeter tells you the total distance around the triangle.
- Two triangles can have the same perimeter but different shapes.
- A wire of a certain length bent into a triangle has its length equal to the perimeter.
Practice Problems
- Find the perimeter of a triangle with sides 7 cm, 10 cm, and 12 cm.
- An equilateral triangle has a side of 22 cm. Find the perimeter.
- The perimeter of a triangle is 55 cm. Two sides are 18 cm and 20 cm. Find the third side.
- An isosceles triangle has equal sides of 14 cm and a base of 10 cm. Find the perimeter.
- The perimeter of an equilateral triangle is 63 cm. Find the length of each side.
- A triangular park has sides 150 m, 200 m, and 250 m. Find the cost of fencing at Rs 25 per metre.
- A wire 72 cm long is bent into an isosceles triangle with a base of 24 cm. Find the length of each equal side.
- Find the perimeter of a triangle whose sides are 3.5 cm, 4.8 cm, and 5.7 cm.
Frequently Asked Questions
Q1. What is the perimeter of a triangle?
The perimeter of a triangle is the total length of all three sides added together. If the sides are a, b, and c, then perimeter = a + b + c.
Q2. What is the perimeter formula for an equilateral triangle?
An equilateral triangle has all three sides equal. If each side is a, the perimeter = 3a. For example, if each side is 10 cm, the perimeter = 30 cm.
Q3. How is perimeter different from area?
Perimeter is the distance around the triangle (measured in cm, m). Area is the space inside the triangle (measured in cm², m²). Perimeter measures length; area measures surface.
Q4. Can two triangles have the same perimeter but different shapes?
Yes. A triangle with sides 10, 10, 10 cm and a triangle with sides 8, 12, 10 cm both have perimeter 30 cm but are very different shapes. One is equilateral, the other is scalene.
Q5. How do you find a missing side if the perimeter is given?
Subtract the sum of the known sides from the perimeter. Missing side = Perimeter − (side 1 + side 2). For example, if P = 40 cm and two sides are 12 cm and 15 cm, the third side = 40 − 12 − 15 = 13 cm.
Q6. What are the units of perimeter?
Perimeter is measured in units of length — cm, m, km, inches, feet, etc. It is NOT measured in square units (that is for area). All sides must be in the same unit before adding.
Q7. If a wire is bent into a triangle, what is the relationship?
The length of the wire equals the perimeter of the triangle it forms. If a 48 cm wire is bent into an equilateral triangle, each side = 48/3 = 16 cm.
Q8. What is the perimeter of a right triangle?
The perimeter of a right triangle is the sum of all three sides, just like any other triangle. Add the two shorter sides (legs) and the longest side (hypotenuse). For example, a right triangle with sides 3, 4, 5 cm has perimeter = 3 + 4 + 5 = 12 cm.
