Triangle Calculator

Side a

Side b

Side c

Result

Area6
Perimeter12
Angle A, °36.8699
Angle B, °53.1301
Angle C, °90
Height to side a4
Inradius1
Circumradius2.5
Triangle typeпрямоугольный

Three sides fix a triangle completely

Area by Heron's formula, all three angles, the height and the inradius and circumradius, with a check that the triangle can exist.

Three sides determine a triangle uniquely — the side-side-side congruence rule. Everything else follows: area from Heron's formula, angles from the law of cosines, heights from the area.

The first thing checked is the triangle inequality: any two sides must sum to more than the third. Sides of 1, 2 and 5 do not form a triangle, and you get an explanation rather than a meaningless number.

What this calculator shows

Area, perimeter and all three angles in degrees
The height to the first side plus the inradius and circumradius
Whether the triangle is acute, right-angled or obtuse

What to keep in mind

Heron's formula works for any triangle, not only right-angled ones — Pythagoras is the special case.
Angles come back in degrees to four places; they always sum to 180 within rounding.

FAQs

What is Heron's formula?

Area from the semi-perimeter: S = √(p(p−a)(p−b)(p−c)) where p is half the perimeter. Its virtue is needing neither a height nor an angle.

Why can a triangle fail to exist?

The triangle inequality: any two sides must exceed the third. With 1, 2 and 5 the two short sides simply cannot reach each other.

How do I get angles from sides alone?

The law of cosines: cos A = (b² + c² − a²) / 2bc, then take the arccosine. For a right triangle it collapses into Pythagoras.

What are the two radii for?

The inradius is area over semi-perimeter and appears in tangency problems. The circumradius is abc/4S and connects to the law of sines.

Worked example

The 3-4-5 triangle

Input: Sides 3, 4 and 5

Output: Area 6, angle C exactly 90°

Note: The classic Pythagorean triple: the calculator identifies it as right-angled because 9 + 16 = 25. The inradius works out to exactly 1.

Useful links

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Triangle Calculator

Result

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