📐 Heron's Formula Calculator
Calculate triangle area from three side lengths using Heron's formula
How to Use This Calculator
Enter Three Side Lengths
Input the lengths of all three sides of the triangle (a, b, c). Make sure all values are positive numbers.
Calculate
Click "Calculate Area" to compute the area using Heron's formula.
Review Results
See the calculated area, semiperimeter, perimeter, and step-by-step calculation process.
Understand the Process
View the detailed step-by-step calculation showing how Heron's formula is applied.
Heron's Formula
Area = √[s(s - a)(s - b)(s - c)]
where s = (a + b + c) / 2 (semiperimeter)
Step 1: Calculate the semiperimeter
s = (a + b + c) / 2
Step 2: Calculate (s - a), (s - b), and (s - c)
These are the differences between the semiperimeter and each side.
Step 3: Multiply all four values together
s × (s - a) × (s - b) × (s - c)
Step 4: Take the square root
Area = √[s(s - a)(s - b)(s - c)]
Advantages of Heron's Formula:
- No need to know the height or angles
- Works for any triangle (acute, right, obtuse)
- Only requires three side lengths
- Very useful when height is difficult to measure
About Heron's Formula Calculator
Heron's Formula Calculator finds the area of a triangle when you know all three side lengths. This formula, attributed to Heron of Alexandria, is especially useful because it doesn't require knowing the height or angles of the triangle.
When to Use This Calculator
- Geometry: Calculate triangle area when height is unknown
- Surveying: Find area from measured side lengths
- Construction: Calculate material needed from side measurements
- Education: Learn and practice Heron's formula
- Land Measurement: Calculate triangular plot areas
- Trigonometry: Find area before using other triangle calculations
Why Use Our Calculator?
- ✅ Step-by-Step: Shows detailed calculation process
- ✅ No Height Needed: Works with just three side lengths
- ✅ Universal: Works for any type of triangle
- ✅ Accurate: Precise mathematical calculations
- ✅ Educational: Helps understand Heron's formula
- ✅ Free: No registration required
Key Concepts
- Semiperimeter: Half the perimeter, s = (a + b + c) / 2
- Triangle Inequality: The sum of any two sides must be greater than the third side
- No Height Required: Unlike Area = ½ × base × height, Heron's formula only needs side lengths
- Universal Formula: Works for acute, right, and obtuse triangles
Example
For a triangle with sides a = 5, b = 6, c = 7:
- Step 1: s = (5 + 6 + 7) / 2 = 9
- Step 2: s - a = 4, s - b = 3, s - c = 2
- Step 3: 9 × 4 × 3 × 2 = 216
- Step 4: Area = √216 ≈ 14.697
Frequently Asked Questions
What is Heron's formula?
Heron's formula calculates the area of a triangle using only the three side lengths: Area = √[s(s-a)(s-b)(s-c)], where s = (a+b+c)/2 is the semiperimeter.
Why use Heron's formula instead of base × height?
Heron's formula is useful when you know the side lengths but not the height. It's especially valuable in surveying and construction where measuring height can be difficult.
Does Heron's formula work for all triangles?
Yes! Heron's formula works for any triangle (acute, right, or obtuse) as long as the three side lengths satisfy the triangle inequality theorem.
What is the semiperimeter?
The semiperimeter (s) is half the perimeter: s = (a + b + c) / 2. It's used in Heron's formula and appears in many triangle formulas.
Can I use Heron's formula if I know the angles?
You can, but you still need all three side lengths. If you only know angles, you'd need to use other formulas like Area = ½ab sin(C).
What happens if the triangle inequality fails?
If a + b ≤ c (or similar), the three lengths cannot form a triangle. The calculator will alert you to this error.
Who was Heron?
Heron of Alexandria (c. 10-70 AD) was an ancient Greek mathematician and engineer. He's credited with developing this formula, though some historians believe it may have been known earlier.