Perimeter of a Triangle with Fractions Calculator
Enter each side of your triangle as a whole number, a fraction, or a mixed number. The calculator adds the three sides exactly using least-common-denominator arithmetic, simplifies the result, and shows every step of the working so you can check your own calculations.
How to find the perimeter of a triangle with fractions
The perimeter of any triangle is simply the sum of its three side lengths: P = a + b + c. When the side lengths are fractions or mixed numbers, you cannot add the numerators directly unless the denominators are the same. The standard approach is to find the least common denominator (LCD) of the three denominators, convert each fraction so that it uses the LCD, add the resulting numerators, and then simplify the sum. A mixed number such as 2 and 1/5 is first converted to an improper fraction (11/5) before being added to the other sides. The final answer can be left as an improper fraction or converted back to a mixed number.
Worked example using this calculator
Suppose a triangle has sides of 2 and 1/5, 4/7, and 1 and 7/8. Convert each to an improper fraction: 11/5, 4/7, and 15/8. The LCD of 5, 7, and 8 is 280. Converting: 11/5 = 616/280, 4/7 = 160/280, 15/8 = 525/280. Adding the numerators: 616 + 160 + 525 = 1301. So the perimeter is 1301/280, which simplifies to 4 and 181/280 (since 1301 divided by 280 = 4 remainder 181, and gcd(181, 280) = 1). As a decimal this is approximately 4.646.
Finding a missing side when the perimeter is known
Switch the calculator to "Missing side" mode if you know the perimeter and two of the three sides. Enter the perimeter in the perimeter fields and the two known sides in the side a and side b fields. The calculator subtracts the two known sides from the perimeter using exact fraction arithmetic: missing = P - a - b. This is the inverse of the addition formula and is useful for geometry homework where the perimeter is given and you must find the third side.
Tips for entering fractions and mixed numbers
Each side is entered using three separate fields: the whole-number part (0 for a pure fraction), the numerator, and the denominator. For example, to enter 3 and 2/9, set the whole part to 3, the numerator to 2, and the denominator to 9. For a pure fraction such as 5/12, set the whole part to 0, numerator to 5, and denominator to 12. For an integer such as 4, set the whole part to 4 and the numerator to 0 (any denominator). The calculator handles the conversion to improper fractions internally and always shows a simplified result.
Common fraction addition quick reference
| Denominators | LCD | Example |
|---|---|---|
| 2 and 3 | 6 | 1/2 + 1/3 = 3/6 + 2/6 = 5/6 |
| 2 and 4 | 4 | 1/2 + 1/4 = 2/4 + 1/4 = 3/4 |
| 3 and 4 | 12 | 1/3 + 1/4 = 4/12 + 3/12 = 7/12 |
| 4 and 5 | 20 | 1/4 + 1/5 = 5/20 + 4/20 = 9/20 |
| 5 and 7 | 35 | 1/5 + 1/7 = 7/35 + 5/35 = 12/35 |
| 3, 4, and 6 | 12 | 1/3 + 1/4 + 1/6 = 4/12 + 3/12 + 2/12 = 9/12 = 3/4 |
| 4, 5, and 8 | 40 | 1/4 + 1/5 + 1/8 = 10/40 + 8/40 + 5/40 = 23/40 |
Reference LCD values for the most frequently seen denominators in school problems.
Frequently asked questions
What is the formula for the perimeter of a triangle with fractional sides?
The formula is the same as for any triangle: P = a + b + c. The difference is that adding fractions requires a common denominator. Find the LCD of the three denominators, convert each fraction, add the numerators, and simplify. For mixed numbers, convert to improper fractions first.
How do I convert a mixed number to an improper fraction?
Multiply the whole number by the denominator and add the numerator. For example, 3 and 2/5 becomes (3 x 5 + 2) / 5 = 17/5. This calculator performs the conversion automatically when you fill in the whole, numerator, and denominator fields.
How do I find the least common denominator of three fractions?
Find the LCD of the first two denominators (the smallest number both divide into evenly), then find the LCD of that result with the third denominator. For example, the LCD of 4, 5, and 8 is LCD(LCD(4,5), 8) = LCD(20, 8) = 40. You can also use prime factorizations: for each prime factor, take the highest power that appears in any denominator.
Why do I need to simplify the final fraction?
After adding, the resulting fraction may have a common factor in the numerator and denominator. Dividing both by their greatest common divisor (GCD) gives the simplest equivalent fraction. For example, 12/16 simplifies to 3/4 because GCD(12, 16) = 4.
Can I use this calculator if one or more sides are whole numbers?
Yes. Enter the whole number in the whole-part field and set the numerator to 0. For instance, a side of length 5 is entered as whole = 5, numerator = 0, denominator = 1 (or any positive integer). The calculator treats it as 5/1 and handles the arithmetic correctly.
What does the missing-side mode do?
It calculates the third side of a triangle when you already know the perimeter and two sides. Enter the perimeter in the perimeter fields and the two known sides in the side a and side b fields. The calculator computes: missing side = Perimeter - side a - side b using exact fraction arithmetic.
What units does the perimeter use?
The calculator is unit-agnostic. The perimeter is expressed in the same units as the side lengths you enter. If the sides are in inches, the perimeter is in inches; if in centimetres, the perimeter is in centimetres.