Triangle Calculator

Written byOleh Soloninupdated 5 October 2026

What you know

Side a (or leg)

Side b (or leg)

Side c

Result

Area6
Workingp = (3 + 4 + 5) ÷ 2 = 6; S = √(6 · 3 · 2 · 1) = 6
Side c5
Perimeter12
Angle A, °36.8699
Angle B, °53.1301
Angle C, °90
Height to side a4
Inradius1
Circumradius2.5
Triangle typeright-angled

Four ways to the area, with the working shown

The area of a triangle from its base and height, three sides (Heron's formula), two sides and the included angle, or the legs of a right triangle — with the working, angles and radii.

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.

How to use

Choose what you know: base and height, three sides, two sides and the angle between them, or the legs of a right triangle.
Enter the numbers in one unit; the area comes out in that unit squared.
Read the area and the working line, and for sides the angles, perimeter and radii as well.

Which measurements are enough

Three sides fix a triangle completely, and Heron's formula turns them into an area without any angle: sides 3, 4 and 5 give an area of 6. A base and a height do the same job more directly, halving their product.

Two sides and the angle between them also suffice, through half the product times the sine of that angle. What never suffices is three angles: they fix the shape but leave the size open, which is what similarity means.

Worked out with this calculator.
Known	Formula	Example	Area
Base and height	½ × b × h	6 and 4	12
Three sides	Heron's formula	3, 4, 5	6
Two sides and the angle	½ × a × b × sin C	enter the angle in the form	—

What this calculator shows

The area with the formula filled in, whichever measurements you have
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.

What is the area of a 3-4-5 triangle?

Six. The sides form a right angle, so half the product of the legs gives it directly; Heron's formula reaches the same number from the three sides alone.

Are three angles enough to find the area?

No. Angles fix the shape but not the size — every similar triangle shares them. At least one length is needed.

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

All tools in Math & Statistics

Triangle Calculator

Result

Related tools

Pythagorean Theorem Calculator

Trapezoid Calculator

Circle Calculator

Useful links