Triangle Area Using Heron's Formula (3 Sides)
Heron's Formula calculates the area of any triangle when all three side lengths are known — no angles, no height required. Given sides a, b, and c, first compute the semi-perimeter s = (a + b + c) / 2, then apply A = √[s(s−a)(s−b)(s−c)]. This approach works for scalene, isosceles, equilateral, and right triangles alike. It is essential in surveying, architecture, engineering, and geometry courses whenever only side measurements are available.
Heron's Formula gives the area of any triangle from its three sides (a, b, c) — no height needed. Step 1: compute the semi-perimeter s = (a+b+c)/2. Step 2: A = √[s·(s−a)·(s−b)·(s−c)]. Example: sides 3, 4, 5 → s=6, A = √(6·3·2·1) = √36 = 6 cm².
When to use this calculator
- Surveying a triangular land plot using only measured boundary distances (no GPS angles) to calculate its area for property registration.
- Verifying the area of a structural triangular steel brace in civil engineering when only the three member lengths are known from blueprints.
- Solving high school and college geometry problems where the triangle's altitude is not given but all three side lengths are provided.
- Calculating the surface area of a triangular sail or fabric panel in manufacturing to estimate material cost per unit.
Worked Example: 3-4-5 right triangle
- a=3, b=4, c=5
- s = (3+4+5)/2 = 6
- A = √(6·3·2·1) = √36 = 6 cm²
How it works
2 min readHow It Is Calculated
Heron's Formula is a two-step process:
Step 1 — Semi-perimeter:
s = (a + b + c) / 2Step 2 — Area:
A = √[ s × (s − a) × (s − b) × (s − c) ]Where a, b, and c are the three side lengths, and s is the semi-perimeter. The result A is in squared units matching the input units (cm → cm², ft → ft², etc.).
> Perimeter is also returned: P = a + b + c = 2s
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Reference Table — Common Triangle Types
The table below shows pre-computed results for common triangle configurations:
| Type | Side a | Side b | Side c | Semi-perm s | Area A |
|---|---|---|---|---|---|
| Right (3-4-5) | 3 | 4 | 5 | 6 | 6.000 |
| Right (5-12-13) | 5 | 12 | 13 | 15 | 30.000 |
| Equilateral (side 6) | 6 | 6 | 6 | 9 | 15.588 |
| Isosceles | 5 | 5 | 6 | 8 | 12.000 |
| Isosceles | 10 | 10 | 12 | 16 | 48.000 |
| Scalene | 7 | 8 | 9 | 12 | 26.833 |
| Scalene | 13 | 14 | 15 | 21 | 84.000 |
| Obtuse | 4 | 6 | 8 | 9 | 7.937 |
| Land plot 15-20-25 m | 15 | 20 | 25 | 30 | 150.000 m² |
All values rounded to 3 decimal places. Units are the square of whichever unit is entered.
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Typical Cases
Case 1 — Classic Right Triangle (3-4-5)
Case 2 — Equilateral Triangle (side = 6 m)
Case 3 — Scalene Triangle (13-14-15 ft)
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Common Errors
1. Forgetting to take the square root — The formula under the radical, s(s−a)(s−b)(s−c), is not the area itself; the area is its square root.
2. Using full perimeter instead of semi-perimeter — s must be half the perimeter. Using P instead of s gives a completely wrong intermediate value.
3. Entering sides that violate the triangle inequality — For a valid triangle, the sum of any two sides must strictly exceed the third (a + b > c). Inputs like 1, 2, 10 produce a negative value under the radical. This calculator will flag such cases.
4. Mixing units — All three sides must be in the same unit before applying the formula. Convert everything first.
5. Rounding sides before computing — Rounding a = 4.997 to 5 before plugging in can shift the area by several percent for near-degenerate triangles.
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Frequently asked questions
What is Heron's Formula and who invented it?
Heron's Formula is attributed to Heron of Alexandria (c. 10–70 AD), a Greek mathematician and engineer. It computes a triangle's area from its three side lengths alone using A = √[s(s−a)(s−b)(s−c)], where s is the semi-perimeter. The formula was documented in his work Metrica and independently known in Indian mathematics via Brahmagupta's generalization (628 AD) for cyclic quadrilaterals.
Does Heron's Formula work for all types of triangles?
Yes — it applies to scalene, isosceles, equilateral, right, acute, and obtuse triangles without exception, as long as the three sides form a valid triangle (triangle inequality: a + b > c for every combination). The only case where it produces zero is a degenerate triangle (collinear points).
How do I know if my three sides form a valid triangle?
Apply the triangle inequality theorem: each side must be strictly less than the sum of the other two. Check all three conditions: a < b + c, b < a + c, and c < a + b. If any condition fails, no triangle exists. For example, sides 2, 3, 10 fail because 2 + 3 = 5 < 10. This calculator automatically detects and reports invalid inputs.
What units does the area come out in?
The area is expressed in the square of whatever unit you enter for the sides. If sides are in centimeters, area is in cm². If sides are in feet, area is in ft². There is no built-in unit conversion — ensure all three sides are entered in the same unit. 1 ft² = 144 in² = 0.0929 m² for reference.
How accurate is Heron's Formula for very flat triangles?
Heron's Formula is mathematically exact, but floating-point implementations can lose precision for near-degenerate triangles (e.g., sides 10, 10, 0.001) because the subtraction s − c approaches zero (catastrophic cancellation). A numerically stable alternative is Kahan's formulation: A = (1/4)√[(a+(b+c))(c−(a−b))(c+(a−b))(a+(b−c))], referenced by NIST.
Can I use Heron's Formula if I know two sides and an angle instead?
No — Heron's Formula strictly requires all three side lengths. If you have two sides and an included angle (SAS), use A = (1/2) × a × b × sin(C) instead. If you have one side and two angles (ASA or AAS), first resolve the third side with the Law of Sines, then apply Heron's.
Why does the 3-4-5 right triangle produce exactly 6 square units?
The 3-4-5 triangle is a Pythagorean triple (3² + 4² = 5²), confirming it is a right angle at the vertex between sides 3 and 4. Its area by the right-triangle formula is A = (1/2) × 3 × 4 = 6. Heron's Formula independently confirms this: s = 6, A = √(6·3·2·1) = √36 = 6. Integer results like this are used in textbooks to validate the formula by hand.
What is the largest area triangle for a fixed perimeter?
For any fixed perimeter P, the equilateral triangle maximizes the area. This follows from the AM-GM inequality applied to Heron's Formula: s(s−a)(s−b)(s−c) is maximized when a = b = c. For P = 12 cm (s = 6), an equilateral triangle of side 4 gives A = √(6·2·2·2) ≈ 6.928 cm², which exceeds any other triangle with the same perimeter.
Is Heron's Formula used in professional surveying?
Yes. Land surveyors routinely decompose irregular polygonal parcels into triangles, measure all side distances, and apply Heron's Formula to compute each sub-area. Results are summed for the total parcel area. This method is codified in standard surveying practices referenced by NIST and state land boards.
How can I find a triangle's height from Heron's Formula?
Once area A is computed, the height h relative to any side b is: h = 2A / b. For the 3-4-5 triangle with A = 6 cm², the height relative to the hypotenuse (side 5) is h = 2×6/5 = 2.4 cm. This is useful for further calculations like centroids or apothem of polygons.