Perimeter of a Triangle
Find the perimeter of any triangle: add three sides, use Pythagoras for a right triangle's missing side, and handle isosceles and equilateral shortcuts.
Perimeter of a Triangle: The One Formula That Covers Every Case
The perimeter of a triangle is the sum of its three side lengths. For any triangle, with sides a, b, and c, the rule is a + b + c. This works for right triangles, isosceles triangles, equilateral triangles, and every irregular triangle in between. The only extra step you need is when a side is missing: in a right triangle, use the Pythagorean theorem to find it first; in any triangle, check the triangle inequality to confirm the three sides can actually form a closed shape.
Triangle Perimeter Formula: a + b + c
Write the three side lengths in the same unit: all feet, all inches, all meters. Then add them. If a triangle has sides of 5 cm, 7 cm, and 9 cm, its perimeter is 5 + 7 + 9 = 21 cm. That is the entire formula. The most common failure is mixing units: a side listed as 2 feet and another as 8 inches must be converted to the same unit before adding. Convert everything to inches (24 + 8 = 32 inches) or to feet (2 + 0.667 = 2.667 feet).
Perimeter of an Equilateral Triangle: The Shortcut
An equilateral triangle has three equal sides. Measure one side and multiply by 3. A triangle with side length 6 inches has perimeter 6 × 3 = 18 inches. This is the fastest triangle perimeter shortcut and the one taught at Grade 3 level (Common Core 3.MD.D.8).
Perimeter of an Isosceles Triangle: Two Equal Sides
An isosceles triangle has two equal sides and one base. Measure one of the equal sides, double it, then add the base. If the equal legs are 8 cm each and the base is 5 cm, the perimeter is (2 × 8) + 5 = 21 cm. This works whether the equal sides meet at the vertex or form the base itself; the shortcut is the same.
Perimeter of a Right Triangle: Find the Missing Side First
When you have two sides of a right triangle but not the third, use the Pythagorean theorem (a² + b² = c², where c is the hypotenuse). This standard is 8.G.B.7 from the Common Core. If the two legs are 3 m and 4 m, the hypotenuse is √(3² + 4²) = √25 = 5 m. Then add all three: 3 + 4 + 5 = 12 m. If you are missing a leg instead of the hypotenuse, rearrange: leg = √(c² − other leg²). A triangle with hypotenuse 13 and one leg 5 has the other leg equal to √(13² − 5²) = √(169 − 25) = √144 = 12.
Check the Triangle Inequality Before You Trust the Perimeter
Three side lengths only form a triangle if the sum of the two shorter sides exceeds the longest side. This is the triangle inequality. Sides of 2, 3, and 6 cannot make a triangle because 2 + 3 = 5, which is less than 6. The perimeter you calculate from those numbers is meaningless; no closed shape exists. Always test this before using the triangle perimeter rule on real-world measurements, especially when one side is missing and you have estimated it.
Worked Examples: Three Common Triangle Perimeter Problems
Example 1: All Three Sides Known
A triangle has sides 12 cm, 15 cm, and 20 cm. Sum: 12 + 15 + 20 = 47 cm. Check the triangle inequality: 12 + 15 = 27, which is greater than 20. The perimeter is 47 cm.Example 2: Equilateral With One Side Given
A triangle with all sides 7 inches. Perimeter: 7 × 3 = 21 inches. No inequality check needed: the sides are equal, so any two sum to double the third, always greater than the third.Example 3: Right Triangle With a Missing Hypotenuse
A right triangle has legs 9 m and 12 m. Hypotenuse: √(9² + 12²) = √(81 + 144) = √225 = 15 m. Perimeter: 9 + 12 + 15 = 36 m. Triangle inequality: 9 + 12 = 21 > 15, 9 + 15 = 24 > 12, 12 + 15 = 27 > 9. The shape is valid.What Goes Wrong Most Often With Triangle Perimeter
The single most common mistake is confusing perimeter with area. Perimeter is a linear measurement: the distance around the boundary, not the space inside. Adding the sides is correct; multiplying them is not. The second most common failure is skipping the triangle inequality check when a side is missing and has been calculated. If a student computes a missing side of 18 for a triangle with two known sides of 5 and 12, the sum 5 + 12 equals 17, which is less than 18; no triangle exists, and the perimeter is zero.
| Triangle Type | Perimeter Formula | Example (Sides) | Perimeter |
|---|---|---|---|
| Equilateral | 3 × side | 5 cm, 5 cm, 5 cm | 15 cm |
| Isosceles | (2 × equal side) + base | 8 m, 8 m, 5 m | 21 m |
| Right | a + b + c (c found via Pythagorean theorem) | 3 ft, 4 ft, 5 ft | 12 ft |
| Scalene (irregular) | a + b + c | 7 in, 9 in, 11 in | 27 in |
Common Questions
What is the perimeter of a triangle?
The perimeter of a triangle is the total distance around it, found by adding its three side lengths: a + b + c.
How do I find the perimeter of a right triangle when only two sides are given?
Use the Pythagorean theorem (a² + b² = c²) to find the missing side, then add all three sides together.
What is the triangle inequality and why does it matter for perimeter?
The triangle inequality says the sum of any two sides must exceed the third. If it fails, those side lengths cannot form a triangle, so the perimeter does not exist.
Can I use the same formula for an equilateral and an isosceles triangle?
Yes, a + b + c works for any triangle. Equilaterals and isosceles have shortcuts: multiply one side by 3 or double the equal sides and add the base, but the standard rule always applies.