3-Sides Triangle Area Calculator — Heron's Formula
Know all three side lengths but not the height? Enter them here and get the triangle's area instantly via Heron's formula — along with all three angles, three altitudes, the inradius and the circumradius.
Computed with Heron's formula from all three side lengths
- 1
Semi-perimeter s = (a + b + c) ÷ 2
(3 + 4 + 5) ÷ 2 = 6 - 2
Inner product s(s−a)(s−b)(s−c)
6 × 3 × 2 × 1 = 36 - 3
Area = √(inner product)
√36 = 6Heron's formula — no height measurement needed.
How does this calculator work?
Use Heron's formula: s = (a+b+c)/2, then Area = √(s(s−a)(s−b)(s−c)). Enter the three side lengths to get the area, all three angles (via the Law of Cosines), the three altitudes, inradius and circumradius. No height measurement needed.
Formula
How this is calculated
Heron's formula computes a triangle's area directly from its three side lengths without needing the height. First calculate the semi-perimeter s = (a + b + c) / 2, then Area = √(s(s − a)(s − b)(s − c)). The formula is algebraically equivalent to the standard ½ × base × height, but avoids having to calculate or measure the perpendicular height.
Once the area is known, all three interior angles follow from the Law of Cosines: cos A = (b² + c² − a²) / (2bc), and similarly for B and C. The angles always sum to exactly 180°. Each altitude (perpendicular height from a vertex) is h_a = 2 × Area / a, and so on for b and c.
The inradius (radius of the inscribed circle) is r = Area / s, and the circumradius (radius of the circumscribed circle) is R = abc / (4 × Area). The calculator rejects inputs that violate the triangle inequality (a + b ≤ c, etc.) because no valid triangle can be formed — the formula would return an imaginary result.
Frequently asked questions
Heron's formula gives the area of a triangle from its three side lengths: A = √(s(s−a)(s−b)(s−c)), where s = (a+b+c)/2 is the semi-perimeter. It works for any triangle — acute, obtuse or right-angled — without needing the height.
The three sides must satisfy the triangle inequality: each side must be less than the sum of the other two. For example, sides 1, 2, 10 are impossible — no triangle can be formed. The calculator checks this and warns you.
Using the Law of Cosines: angle A = arccos((b² + c² − a²) / (2bc)), where a is the side opposite angle A. The same formula is applied for B and C, and the three results should sum to 180°.
Also known as
TG we-Calculate Editorial Team. (2026). 3-Sides Triangle Area Calculator — Heron's Formula [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/3-sides-triangle-area-calculator
TG we-Calculate Editorial Team. "3-Sides Triangle Area Calculator — Heron's Formula." TG we-Calculate. 2026. https://we-calculate.com/calculator/3-sides-triangle-area-calculator.
TG we-Calculate Editorial Team, "3-Sides Triangle Area Calculator — Heron's Formula," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/3-sides-triangle-area-calculator
@misc{wecalculate_3_sides_triangle_area_calculator, title = {3-Sides Triangle Area Calculator — Heron's Formula}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/3-sides-triangle-area-calculator}}, year = {2026}, note = {TG we-Calculate} }
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