Calculator Tools
Triangle Calculator
Solve any triangle from three known parts. Five modes (SSS, SAS, ASA, AAS, SSA) with sides, angles, area, perimeter, altitudes, inradius, and circumradius.
Triangle calculator
Pick a solving mode
Sides accept integers, decimals, and simple fractions like 3/4. Angles use the unit selector on the right.
Enter all three side lengths. Angles are solved by the law of cosines.
Solution
Area = 6
Right scalene. Perimeter = 12.
Sides
Each side is opposite the angle of the same letter.
Side a
3
Side b
4
Side c
5
Angles
Switch between degrees and radians with the unit selector above.
Angle A
36.869898 deg
Opposite side a
Angle B
53.130102 deg
Opposite side b
Angle C
90 deg
Opposite side c
Perimeter and area
Area uses Heron's formula sqrt(s(s-a)(s-b)(s-c)) where s is the semi-perimeter.
Perimeter
12
a + b + c
Semi-perimeter (s)
6
(a + b + c) / 2
Area
6
Heron's formula
Inscribed and circumscribed circles
Inradius r = area / s. Circumradius R = (a b c) / (4 area).
Inradius (r)
1
Radius of the inscribed circle
Circumradius (R)
2.5
Radius of the circumscribed circle
Triangle type
Right scalene
Classified by largest angle and by side equality
Altitudes (heights)
Each altitude is the perpendicular distance from a vertex to the opposite side. h_x = 2 * area / x.
h_a (height to side a)
4
h_b (height to side b)
3
h_c (height to side c)
2.4
Step-by-step
Each line shows the substitution used to reach the next result.
- Inputs (SSS): a = 3, b = 4, c = 5.
- Triangle inequality holds: a + b > c (7 > 5), a + c > b, and b + c > a.
- Angle A by law of cosines: cos(A) = (b^2 + c^2 - a^2) / (2 b c) = (16 + 25 - 9) / 40 = 0.8. So A = acos(0.8) = 36.869898 deg.
- Angle B by law of cosines: cos(B) = (a^2 + c^2 - b^2) / (2 a c) = 0.6. So B = acos(0.6) = 53.130102 deg.
- Angle C closes the triangle: C = 180 - A - B = 90 deg.
- Semi-perimeter s = (a + b + c) / 2 = 6.
- Area by Heron: sqrt(s(s-a)(s-b)(s-c)) = sqrt(6 * 3 * 2 * 1) = 6.
- Altitudes (h = 2 * area / side): h_a = 4, h_b = 3, h_c = 2.4.
- Inradius r = area / s = 1. Circumradius R = (a b c) / (4 area) = 2.5.
Law of sines and law of cosines
a / sin(A) = b / sin(B) = c / sin(C) = 2 R
c^2 = a^2 + b^2 - 2 a b cos(C)
- The law of sines relates each side to the sine of its opposite angle. It is the right tool when an angle and its opposite side are paired.
- The law of cosines is a generalization of the Pythagorean theorem and is the right tool for SSS and SAS, where no side-angle pair is known up front.
- R, the circumradius, is the common ratio in the law of sines. The inradius r equals area divided by the semi-perimeter.
Which mode should I pick?
- SSS: all three side lengths are known. Always unambiguous when the triangle inequality is satisfied.
- SAS: two sides and the angle between them. Always unambiguous.
- ASA: two angles and the side between them. Always unambiguous.
- AAS: two angles and a side opposite one of them. Always unambiguous.
- SSA: two sides and an angle that is not between them. This is the ambiguous case and can give 0, 1, or 2 valid triangles.
How to use
- Pick a solving mode at the top: SSS, SAS, ASA, AAS, or SSA. The form below switches its three fields to match the chosen mode.
- Toggle the angle unit between degrees and radians if needed. Sides accept integers, decimals, comma decimals, and simple fractions such as 3/4.
- Type the three known values into the labeled fields. Tap any preset (3-4-5, ambiguous SSA, and more) to seed a familiar example in one click.
- Read the headline solution: area, perimeter, and triangle classification. If SSA produced two triangles, both are shown one after the other with a note.
- Open the Sides, Angles, Perimeter, Inscribed/Circumscribed, and Altitudes panels to see each computed value with one-click copy buttons.
- Open the Step-by-step panel to see each law-of-sines or law-of-cosines substitution. Use Copy full report or Copy steps to paste into homework, notes, or a write-up.
About this tool
Triangle Calculator solves any triangle (acute, right, or obtuse) given three of its parts using the law of sines and the law of cosines. Pick one of five standard modes: SSS when all three sides are known and angles must be recovered; SAS when two sides and the angle between them are known; ASA when two angles and the side between them are known; AAS when two angles and a non-included side are known; or SSA, the classic ambiguous case, when two sides and a non-included angle are known. SSA can produce zero, one, or two valid triangles depending on the inputs, and the calculator reports every valid solution side by side with a note explaining which case applies. Each solved triangle reports all three sides, all three angles (in degrees and radians via the unit toggle), perimeter, semi-perimeter, area by Heron's formula, the three altitudes (heights to each side), the inradius of the inscribed circle, the circumradius of the circumscribed circle, and a classification of the triangle by largest angle (acute, right, obtuse) and by side equality (equilateral, isosceles, scalene). A step-by-step derivation mirrors the substitution work expected on a homework page, so the result is not just a number but a transparent argument you can paste into notes. Inputs accept integers, decimals, comma decimals used in European locales, and simple fractions such as 3/4 for everything from textbook drills to surveying numbers. The triangle inequality, angle-sum constraint, and SSA branch logic are validated explicitly so impossible inputs produce a clear message rather than a silent NaN. Computation uses native Math primitives in your browser, so the lengths and angles you enter never leave your device, and there is no signup, no upload, and no analytics on the result. Useful for trigonometry homework, geometry coursework, surveying and land plots, construction layout, engineering drawings, navigation problems, physics force triangles, and any time a triangle has to be closed from a few measured parts.
Free to use. Works in your browser. No signup, no login.
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