Triangle Calculator
From three side lengths get the area (Heron's formula), all three angles, perimeter, heights, inradius and circumradius, and whether it's right, acute or obtuse.
Triangle Calculator: with the default inputs, area is 14.6969 m².
- Type
- Scalene, acute
- Perimeter
- 18 m
- Angle A (opposite a)
- 44.42 deg
- Angle B (opposite b)
- 57.12 deg
- Angle C (opposite c)
- 78.46 deg
- Height to side a
- 5.8788 m
- Height to side b
- 4.899 m
- Height to side c
- 4.1991 m
- Inradius (inscribed circle)
- 1.633 m
- Circumradius (circumscribed circle)
- 3.5722 m
Assumptions
- Worked steps are shown in meters (the internal base unit); results are converted to the units you choose.
- Right/equal-side classification uses a relative tolerance of 10⁻⁹ to absorb floating-point noise.
How this is worked out
The formula
s = (a + b + c) ÷ 2 Area = √( s (s − a) (s − b) (s − c) ) (Heron's formula) cos A = (b² + c² − a²) ÷ 2bc (law of cosines; likewise for B; C = 180° − A − B) Height to side a: h_a = 2 × Area ÷ a Inradius r = Area ÷ s, Circumradius R = abc ÷ 4·Area
Open How it’s calculated above to see this worked through with your own numbers.
What you enter
- Side a
- A number.in mm, cm, m, km, in, ft, yd, mi · 0 or more · defaults to 5
- Side b
- A number.in mm, cm, m, km, in, ft, yd, mi · 0 or more · defaults to 6
- Side c
- A number.in mm, cm, m, km, in, ft, yd, mi · 0 or more · defaults to 7
What you get back
- Areamain answer
- Type
- Perimeter
- Angle A (opposite a)
- Angle B (opposite b)
- Angle C (opposite c)
- Height to side a
- Height to side b
- Height to side c
- Inradius (inscribed circle)
- Circumradius (circumscribed circle)
What this assumes
- Worked steps are shown in meters (the internal base unit); results are converted to the units you choose.
- Right/equal-side classification uses a relative tolerance of 10⁻⁹ to absorb floating-point noise.
About this calculator
Three side lengths pin down a triangle completely — its shape, its angles, its area. (Three angles don't: they fix the shape but not the size.) This calculator takes the three sides and returns everything else a geometry problem might ask for, and it checks first that the triangle can exist.
How to use it
Enter sides a, b and c in any unit. Angle A is opposite side a, B opposite b, C opposite c — the standard labelling. Switch the angle unit to radians if you need them for a formula, and the area unit to whatever fits the problem.
Heron's formula
You don't need a height to find a triangle's area. With the semi-perimeter s = (a + b + c)/2, the area is √(s(s − a)(s − b)(s − c)). It looks odd, but it's just ½ × base × height with the height eliminated algebraically using Pythagoras. Once you have the area you can recover any height: h = 2 × area ÷ base.
The angles
The law of cosines, c² = a² + b² − 2ab·cos C, is the Pythagorean theorem with a correction term for non-right angles. Rearranged, it gives each angle directly from the three sides. The calculator computes two angles that way and gets the third from the fact that they must sum to 180°, which also serves as a check.
Triangle inequality
The longest side must be shorter than the other two combined. If it's equal, the "triangle" is a flat line with zero area; if it's longer, the two short sides can't reach each other. The calculator refuses both cases rather than returning NaN.
Classifying the triangle
- By sides: equilateral (all equal), isosceles (two equal), scalene (none equal).
- By angles: compare the square of the longest side with the sum of the squares of the other two. Equal → right; smaller → acute; larger → obtuse. This is the converse of the Pythagorean theorem.
Inradius and circumradius
The inradius is the radius of the largest circle that fits inside (area ÷ s). The circumradius is the radius of the circle through all three corners (abc ÷ 4·area). Both come up in construction and in competition geometry.
Frequently asked questions
▸How do I find the area of a triangle with three sides?
Use Heron's formula: find s = (a + b + c)/2, then area = √(s(s − a)(s − b)(s − c)). For 5, 6, 7: s = 9 and area = √(9 × 4 × 3 × 2) = √216 ≈ 14.70.
▸How do I find the angles from the sides?
Use the law of cosines: cos A = (b² + c² − a²) ÷ 2bc, then take the inverse cosine. Repeat for B, and C = 180° − A − B.
▸How do I know if three lengths can form a triangle?
The longest side must be shorter than the sum of the other two. 3, 4, 8 fails because 8 > 3 + 4; 3, 4, 5 works.
▸How can I tell if a triangle is right, acute or obtuse from its sides?
Compare the longest side squared with the sum of the other two squared. Equal is right (25 = 9 + 16), less is acute, more is obtuse.
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