Mathwords logoMathwords

Semiperimeter

Semiperimeter

Half the perimeter of a plane figure.

 

 

See also

Heron’s formula, area of a triangle

Key Formula

s=a+b+c2s = \frac{a + b + c}{2}
Where:
  • ss = The semiperimeter of the triangle
  • aa = Length of the first side
  • bb = Length of the second side
  • cc = Length of the third side

Worked Example

Problem: Find the semiperimeter of a triangle with sides 5, 12, and 13.
Step 1: Add up all three side lengths to find the perimeter.
P=5+12+13=30P = 5 + 12 + 13 = 30
Step 2: Divide the perimeter by 2 to get the semiperimeter.
s=302=15s = \frac{30}{2} = 15
Step 3: Verify: the semiperimeter should be greater than each individual side. Here 15 > 13, 15 > 12, and 15 > 5. ✓
Answer: The semiperimeter is 15.

Another Example

This example shows the main reason students need the semiperimeter: it feeds directly into Heron's formula to find a triangle's area without knowing the height.

Problem: A triangle has sides 7, 8, and 9. Use the semiperimeter and Heron's formula to find its area.
Step 1: Calculate the semiperimeter.
s=7+8+92=242=12s = \frac{7 + 8 + 9}{2} = \frac{24}{2} = 12
Step 2: Compute each factor needed for Heron's formula: (s − a), (s − b), and (s − c).
s−a=12−7=5,s−b=12−8=4,s−c=12−9=3s - a = 12 - 7 = 5, \quad s - b = 12 - 8 = 4, \quad s - c = 12 - 9 = 3
Step 3: Substitute into Heron's formula.
A=s(s−a)(s−b)(s−c)=12×5×4×3A = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{12 \times 5 \times 4 \times 3}
Step 4: Simplify the product under the square root and evaluate.
A=720=144×5=125≈26.83A = \sqrt{720} = \sqrt{144 \times 5} = 12\sqrt{5} \approx 26.83
Answer: The area of the triangle is 12√5 ≈ 26.83 square units.

Frequently Asked Questions

Why do you use semiperimeter instead of just the perimeter?
The semiperimeter simplifies several important formulas. In Heron's formula, using s instead of P/2 everywhere makes the expression much cleaner and easier to work with. It also appears naturally in the formulas for the inradius and circumradius of a triangle.
Does semiperimeter apply only to triangles?
No. Any plane figure has a semiperimeter — it is simply half the perimeter. For a rectangle with length l and width w, the semiperimeter is l + w. However, the semiperimeter is most commonly used with triangles because of Heron's formula and related triangle properties.
What is the relationship between semiperimeter and inradius?
For any triangle, the area equals the inradius times the semiperimeter: A = r × s. This means if you know the area and the semiperimeter, you can find the inradius by r = A/s. This relationship connects the triangle's inscribed circle to its side lengths.

Semiperimeter vs. Perimeter

SemiperimeterPerimeter
DefinitionHalf the total distance around a figureThe total distance around a figure
Formula (triangle)s = (a + b + c) / 2P = a + b + c
Relationships = P / 2P = 2s
Primary useHeron's formula, inradius, and other triangle formulasMeasuring total boundary length of any shape

Why It Matters

You encounter the semiperimeter most often in Heron's formula, which lets you calculate a triangle's area when you know only the three side lengths — no height required. It also appears in the inradius formula (r = A/s) and in Brahmagupta's formula for cyclic quadrilaterals. Competitive math problems and standardized tests frequently use it, so recognizing when to compute the semiperimeter saves you significant time.

Common Mistakes

Mistake: Forgetting to divide by 2 and using the full perimeter in Heron's formula.
Correction: Heron's formula requires the semiperimeter s = (a + b + c)/2, not the perimeter. Using the full perimeter produces an answer that is too large by a factor of 4 under the radical.
Mistake: Computing (s − a), (s − b), (s − c) incorrectly by subtracting from the perimeter instead of the semiperimeter.
Correction: Each factor must subtract a side from s, not from P. For example, if s = 12 and a = 7, the factor is 12 − 7 = 5, not 24 − 7 = 17.

Related Terms