Heron's Formula: Finding a Triangle's Area From Its Three Sides
October 29, 2026
You can find a triangle's area as half the base times the height, but in practice you often know only the three side lengths: for example, when measuring a triangular garden bed or a corner lot with a tape measure. Heron's formula gives the area directly from the sides a, b, and c. First compute the semi-perimeter, s = (a + b + c) ÷ 2. Then the area is √(s(s − a)(s − b)(s − c)).
For the classic triangle with sides 13, 14, and 15, the semi-perimeter is 21, and the area is √(21 × 8 × 7 × 6) = √7,056 = 84 square units. The formula is attributed to Hero of Alexandria, who lived in the first century.
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Open the Area Calculator →Step by Step
Step 1: measure all three sides in the same unit. Step 2: add them and divide by 2 to get s. Step 3: compute the three differences s − a, s − b, s − c. Step 4: multiply s by the three differences. Step 5: take the square root. For a triangular lot with sides 40, 55, and 70 feet: s = 82.5; differences are 42.5, 27.5, and 12.5; the product is 82.5 × 42.5 × 27.5 × 12.5 = 1,205,273, and the square root is 1,097.8 square feet. The result is in square feet if the sides are in feet.
Check that the sides can form a triangle: the sum of any two sides must exceed the third. If one difference s − a is zero or negative, the 'triangle' is flat or impossible.
| Sides | Semi-perimeter | Area | Note |
|---|---|---|---|
| 13, 14, 15 | 21 | 84 | Classic example |
| 60, 80, 100 | 120 | 2,400 | Right triangle (half of 60 x 80) |
| 40, 55, 70 ft | 82.5 | 1,097.8 sq ft | Triangular lot |
| 120, 150, 200 | 235 | 8,966.6 | Large field |
| 5, 5, 5 | 7.5 | 10.83 | Equilateral |
Checking With the Base-Height Method
For the 60-80-100 right triangle, base 80 and height 60 give area ½ × 80 × 60 = 2,400, matching Heron's formula. If you know a base and the perpendicular height, that method is simpler. If you only have sides, Heron's formula is handy, and for irregular plots you can divide the shape into triangles, measure each side, and add the areas. This is how surveyors with tape measures historically estimated field areas. For a quadrilateral, split it with a diagonal into two triangles, measure the five sides, and apply Heron's formula twice.
Numerical Care
For very thin triangles (one side nearly equal to the sum of the other two), Heron's formula can lose accuracy in floating-point arithmetic because s − a is a small difference of large numbers. Computer implementations use a rearranged version that sorts the sides and combines them in a stable way. For hand calculations with a tape measure, the main source of error is the measurements themselves: a 1% error in each side can produce several percent error in the area for a thin triangle.
Uses
Estimating the area of triangular roof sections or gables for shingles and paint; calculating land for sale or tax; planning garden beds, patios, and sails; and computing the area under a triangular load. Once you have the area, you can estimate materials: for 1,097.8 square feet of lawn, seed at 5 pounds per 1,000 square feet needs about 5.5 pounds. The same formula appears in geometry classes as an example of using algebra to avoid finding the height.
Frequently Asked Questions
What is Heron's formula?
Area = √(s(s − a)(s − b)(s − c)) where s is half the perimeter.
When should I use Heron's formula?
When you know the three side lengths but not the height.
What is the area of a 13-14-15 triangle?
84 square units.
How do I know if three lengths form a triangle?
The sum of any two must be greater than the third.
Can I use it for four-sided plots?
Yes. Divide the plot into two triangles using a diagonal and add their areas.