Triangle Calculator
Solve any triangle from three inputs, three vertex coordinates, or two right-triangle sides. Get every side, angle, area, perimeter, median, altitude, angle bisector, inradius, circumradius and centroid — drawn to scale.
What is a triangle solver?
A triangle solver takes any three known facts about a triangle — sides, angles, or a mix — and finds the missing three. That is the classic geometry problem: once you know just enough, every other measurement in the triangle is locked in. This calculator handles all five solvable cases (SSS, SAS, ASA, AAS, SSA), the tricky ambiguous SSA case where two different triangles can fit the same numbers, plus coordinate geometry and a fast Pythagorean path for right triangles.
Triangles matter far beyond the classroom. Roof pitches, ramps, staircases, bridges, antenna towers, camera tripods, sail plans and even game engines all rely on the same three rules underneath — the Law of Sines, the Law of Cosines, and the sum of angles being 180°. Get comfortable with those three and you can rebuild every triangle formula from scratch.
To use the solver, pick the mode that matches what you know, type the values, and read the results. Angles can be entered in degrees or radians. The step-by-step panel shows exactly which rule was applied, so you can use it as a study tool as well as an answer key.
Triangle solving methods explained, case by case
For every solving case below you'll see a plain-English explanation, the formula (with what each letter means), a diagram showing which parts are known, and a worked example — all in one place.
SSS — three sides
When you know all three sides, the triangle is fully fixed (as long as the triangle inequality holds: any two sides sum to more than the third). Use the Law of Cosines rearranged to solve for each angle in turn.
SAS — two sides and the included angle
Given two sides and the angle between them, the Law of Cosines gives the third side directly. From there the Law of Sines finds the remaining angles.
ASA — two angles and the included side
Two angles determine the third (they sum to 180°), and the given side is between them. The Law of Sines then scales the two missing sides.
AAS — two angles and a non-included side
Same tool as ASA — the third angle is 180° − A − B, then apply the Law of Sines. The difference is only that the given side is opposite one of the given angles, not between them.
SSA — the ambiguous case
Two sides and an angle not between them can leave 0, 1 or 2 valid triangles. The Law of Sines gives sin B = b·sin A / a; if that value exceeds 1, no triangle fits; if it equals 1, one right triangle fits; otherwise two candidate B values (B and 180° − B) must both be checked.
Right triangle (Pythagorean fast path)
If one angle is 90°, Pythagoras alone finds any missing side from the other two. The two acute angles come straight from the ratio of legs to the hypotenuse.
Coordinates — triangle from three vertices
With three (x, y) points, side lengths come from the distance formula, area from the Shoelace formula, and the centroid is just the average of the vertices. Angles then follow via the Law of Cosines.
Types of triangles
By sides:
- Equilateral — all three sides equal (all angles 60°).
- Isosceles — exactly two sides equal (base angles equal).
- Scalene — no sides equal.
By angles:
- Right — one angle equals 90°.
- Obtuse — one angle greater than 90°.
- Acute — all three angles less than 90°.
Special right triangles
| Triangle | Angles | Side ratio | Notes |
|---|---|---|---|
| 30-60-90 | 30°, 60°, 90° | 1 : √3 : 2 | Half of an equilateral triangle. |
| 45-45-90 | 45°, 45°, 90° | 1 : 1 : √2 | Isosceles right triangle — the diagonal of a square. |
Use the "30-60-90 preset" or "45-45-90 preset" buttons above the input grid to auto-fill these.
Cevians and radii — quick reference
Triangle centers
Every triangle has several distinguished points that stay meaningful no matter how you rotate, translate or scale it. This calculator reports four of them: the centroid, incenter, circumcenter and orthocenter.
Centroid (G) — the average of the three vertices, G = (A + B + C) / 3. It's where the three medians meet and is the triangle's balance point: cut the triangle from card and it will balance on a pin placed at G.
Incenter (I) — the intersection of the three angle bisectors, and the center of the inscribed circle (the incircle) that just touches all three sides. In coordinates it is the weighted average I = (a·A + b·B + c·C) / (a + b + c), where the weights are the lengths of the sides opposite each vertex. The incircle has radius r = Area / s.
Circumcenter (O) — the intersection of the three perpendicular bisectors of the sides, and the center of the circumscribed circle (the circumcircle) that passes through all three vertices. Its radius is R = a / (2·sin A). For an acute triangle O lies inside; for a right triangle it sits exactly on the midpoint of the hypotenuse; for an obtuse triangle it lies outside.
Orthocenter (H) — the intersection of the three altitudes (the perpendiculars dropped from each vertex to the opposite side). It's inside acute triangles, at the right-angle vertex of a right triangle, and outside obtuse triangles.
G, O and H are always collinear on a single line called the Euler line, and G divides the segment OH in the ratio 1 : 2 — equivalently, H = 3G − 2O, which is exactly how the orthocenter is derived here from the centroid and circumcenter.
Nine-point circle (N) — a fifth remarkable circle that passes through the midpoint of each side, the foot of each altitude, and the midpoint of each segment from a vertex to the orthocenter (nine points in total). Its center N is the midpoint of O and H (also on the Euler line), and its radius is always half the circumradius, R/2. Toggle it on in the diagram above.
Similarity / congruence checker
Compare two triangles to see if they're similar (same shape) or congruent (same shape and size). Enter each triangle as either its three sides or its three angles.
Triangle 1
Triangle 2
Features of this calculator
- Three modes — General solver (SSS/SAS/ASA/AAS/SSA), Coordinates (three (x,y) vertices), and Right triangle (any 2 of legs a, b or hypotenuse c).
- Detects the ambiguous SSA case and returns both triangles when they exist.
- Live labeled SVG diagram drawn roughly to scale for a fast sanity check.
- Area computed two ways — Heron's formula and ½·a·b·sinC — as an automatic cross-check.
- Every classical cevian: three medians, three altitudes (hₐ = 2·Area/a), and three angle bisector lengths.
- Inradius (Area / s), circumradius (a / 2·sinA) and — in Coordinates mode — the centroid ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3).
- Quick-fill presets for the classic 30-60-90 and 45-45-90 right triangles.
- Similar-triangle scale-factor tool: enter k and get the scaled sides, perimeter (×k) and area (×k²).
- Optional nine-point circle overlay — center N = midpoint(H, O), radius R/2.
- Similarity / congruence checker — tests AA, SAS, SSS (similarity) and SSS, SAS, ASA, AAS, HL (congruence) between two triangles you enter by sides or angles.
- Angle inputs and outputs in degrees or radians via a toggle.
- Show/hide step-by-step working with the specific law used, substitutions and running totals.
Frequently asked questions
+What information do I need to solve a triangle?
Any three values, with at least one side. That covers SSS (three sides), SAS (two sides and the included angle), ASA (two angles and the included side), AAS (two angles and a non-included side) and SSA (two sides and a non-included angle). Three angles alone don't define a unique triangle — only its shape.
+What is the ambiguous SSA case?
When you know two sides and an angle that is NOT between them, there can be 0, 1 or 2 triangles that fit. The calculator detects this automatically and shows both solutions when they exist.
+How is the area computed?
Once all three sides are known, Heron's formula A = √[s(s−a)(s−b)(s−c)] gives the area from the semi-perimeter s. As a cross-check the calculator also shows A = ½·a·b·sin(C) using the two-sides-and-included-angle form.
+What are the median, inradius and circumradius?
A median joins a vertex to the midpoint of the opposite side; each has length mₐ = ½√(2b²+2c²−a²). The inradius r = Area/s is the radius of the inscribed circle. The circumradius R = a / (2 sin A) is the radius of the circle passing through all three vertices.
+Can I enter angles in radians?
Yes — use the Degrees / Radians toggle above the input grid. Results are reported in the same unit.
+Do you support coordinate geometry and right triangles?
Yes. Switch to Coordinates mode to enter three (x, y) vertices — the calculator derives every side, angle, area (via the cross-product / shoelace method), perimeter and centroid. Right triangle mode lets you enter any 2 of legs a, b or hypotenuse c and applies the Pythagorean theorem directly.
+What is the nine-point circle?
It's the circle that passes through nine significant points of a triangle — the midpoint of each side, the foot of each altitude, and the midpoint of each segment from a vertex to the orthocenter. Its center N is the midpoint of the orthocenter H and circumcenter O (it lies on the Euler line), and its radius is exactly half the circumradius, R/2. Toggle it on in the diagram overlays.
+How can I check if two triangles are similar or congruent?
Use the Similarity / Congruence Checker tool below the solver. Enter each triangle as either its three sides or its three angles, and it tests AA, SAS and SSS for similarity, and SSS, SAS, ASA, AAS and HL for congruence, reporting which criterion matches (and the scale factor if the triangles are similar but not congruent).