Free triangle calculator and solver. Enter any three sides or angles (SSS, SAS, ASA, AAS, SSA) to get every side, angle, area, perimeter, height, and circle radius.
Your results will appear here.
Enter your values and press Calculate.
Tell the solver which case you have: SSS (three sides), SAS (two sides and the angle between them), ASA (two angles and the side between them), AAS (two angles and another side), or SSA (two sides and a non-included angle). For SSS the Law of Cosines finds each angle in turn; for SAS it finds the third side first, then the remaining angles; for ASA and AAS the third angle comes from the 180-degree angle sum and the Law of Sines finds the missing sides; for SSA the Law of Sines finds the second angle, and the ambiguous case can produce two valid triangles, both of which are reported. Once all six measurements are known, Heron's formula gives the area from the semiperimeter, each height is twice the area over its base, each median follows the Apollonius formula, the inradius is area over semiperimeter, and the circumradius is abc over four times the area.
Three workhorse formulas solve almost every triangle. The Law of Cosines, c² = a² + b² - 2ab*cos(C), generalizes the Pythagorean theorem to any angle: when the angle is 90 degrees, cos(C) is zero and you get a² + b² = c². Use it when you know three sides (to find angles) or two sides plus the included angle (to find the third side).
The Law of Sines, a/sin(A) = b/sin(B) = c/sin(C), links each side to the sine of its opposite angle. Use it when you know two angles and a side, or two sides and a non-included angle. The common ratio also equals 2R, twice the circumradius, which is why the circumradius appears in every full solution.
For area, the base-times-height formula A = 1/2 * b * h is the starting point, but you rarely know the height. Heron's formula sidesteps it entirely: from the semiperimeter s = (a+b+c)/2, Area = sqrt(s(s-a)(s-b)(s-c)). With two sides and the included angle, A = 1/2 * a * b * sin(C) is faster.
Pick the case that matches what you know: SSS, SAS, ASA, AAS, or SSA. Then fill in exactly the three values that case asks for and leave the rest at zero. Angles are entered in degrees.
The solver returns every side and angle, the area, perimeter, and semiperimeter, all three heights (altitudes), all three medians, the inradius and circumradius, and the triangle's classification, for example Scalene Right. The solver will show your work: each formula is applied with your own numbers substituted, so you can follow every step. In the SSA ambiguous case it also describes the second valid triangle when one exists.
If your values cannot form a triangle, for example three sides that fail the triangle inequality, the solver says so instead of returning nonsense.
SSS means three sides. The Law of Cosines finds each angle, and the last angle is checked against the 180-degree sum. This case always has exactly one solution.
SAS means two sides and the angle between them. The Law of Cosines finds the third side, then the Law of Sines finds the remaining angles. One solution.
ASA means two angles and the side between them; AAS means two angles and any other side. Both give the third angle from 180 minus the other two, then the Law of Sines finds the missing sides. One solution each.
SSA means two sides and an angle that is not between them. The Law of Sines can yield zero, one, or two solutions, which is why this case gets its own section below.
AAA, three angles with no side, is not solvable: it fixes the shape but not the size, so infinitely many similar triangles fit.
SSA is ambiguous because the Law of Sines only determines sin(B), and every sine value (except 1) belongs to two angles: B and 180 minus B. Whether both fit depends on the numbers.
No solution happens when the known side opposite the angle is too short to reach the third vertex: sin(B) comes out greater than 1, which is impossible. One solution happens when sin(B) equals exactly 1, giving a right angle, or when only one of the two candidate angles keeps the angle sum under 180 degrees. Two solutions happen when both candidates fit, giving two genuinely different triangles from the same three numbers.
This solver checks both possibilities and reports the second triangle explicitly, so you never silently get just one of the two answers.
Triangles are classified two ways at once. By sides: equilateral (all three equal, all angles 60 degrees), isosceles (two equal sides and two equal base angles), or scalene (no equal sides). By angles: acute (every angle under 90 degrees), right (one angle exactly 90 degrees), or obtuse (one angle over 90 degrees).
The two systems combine: a 3-4-5 triangle is scalene and right, while an equilateral triangle is always acute. Together, the side-based and angle-based types give every triangle a two-word label, such as Scalene Right. Because the angles always sum to 180 degrees, a triangle can have at most one right or obtuse angle.
A quick test with the sides alone tells you the angle class. Let c be the longest side: if a² + b² = c² the triangle is right, if greater it is acute, and if smaller it is obtuse. Sides 2, 3, 4 give 4 + 9 = 13 < 16, so that triangle is obtuse.
The 3-4-5 triangle also belongs to the family of Pythagorean triples: integer side sets like 5-12-13 and 8-15-17 that satisfy a² + b² = c² exactly.
Take sides a = 3, b = 4, c = 5, the classic SSS case. Angle A comes from the Law of Cosines: cos(A) = (16 + 25 - 9) / (2*4*5) = 32/40 = 0.8, so A = 36.87 degrees. Angle B: cos(B) = (9 + 25 - 16) / (2*3*5) = 18/30 = 0.6, so B = 53.13 degrees. Angle C = 180 - 36.87 - 53.13 = 90 degrees.
The semiperimeter is s = 6, and Heron's formula gives Area = sqrt(6*3*2*1) = 6. The heights, each height also called an altitude, are 2*Area over each side: 4, 3, and 2.4. A related measurement is the angle bisector, which splits an angle into two equal parts; its length follows from the adjacent sides, though this solver focuses on heights and medians. The inradius is Area/s = 1, and the circumradius is abc/(4*Area) = 60/24 = 2.5, which makes sense: the hypotenuse of a right triangle is the diameter of its circumcircle, so R = 5/2.
Classification: scalene by sides, right by angles.
Any three measurements that include at least one side: three sides (SSS), two sides and an angle (SAS or SSA), or two angles and a side (ASA or AAS). Three angles alone are not enough, since they fix the shape but not the size.
Use Heron's formula. From the semiperimeter s = (a+b+c)/2, Area = sqrt(s(s-a)(s-b)(s-c)). For sides 5, 6, 7: s = 9 and Area = sqrt(9*4*3*2) = sqrt(216) = 14.70. If you know two sides and the included angle, A = 1/2 * a * b * sin(C) is simpler.
Exactly 180 degrees, for every triangle in flat Euclidean geometry. If you know two angles, the third is 180 minus their sum: angles of 47 and 83 leave 50 degrees.
No. Two right angles already sum to 180 degrees, leaving nothing for the third angle. A triangle can have at most one right angle and at most one obtuse angle.
Use the triangle inequality: each side must be shorter than the sum of the other two (a+b > c, a+c > b, b+c > a). Sides 3, 4, 8 fail because 3+4 is not greater than 8, while 3, 4, 6 pass all three checks.
When you know two sides and a non-included angle, the Law of Sines can produce zero, one, or two valid triangles. If the computed sin(B) exceeds 1 there is no triangle; if it equals 1 the triangle is right and unique; otherwise check both B and 180-B, since each may give a valid triangle.
Use the Law of Cosines for SSS (finding angles) and SAS (finding the third side). Use the Law of Sines for ASA, AAS, and SSA. Roughly: cosines connect sides to included angles, sines connect sides to opposite angles.
The inradius r is the radius of the largest circle that fits inside the triangle, touching all three sides; r = Area/s. The circumradius R is the radius of the circle through all three vertices; R = abc/(4*Area). For a right triangle, the hypotenuse is the circumcircle's diameter.
If you know the other two angles, subtract from 180. If you know sides instead, use the Law of Cosines (SSS) or the Law of Sines (SAS, ASA, AAS, SSA) to find the angle directly.
The largest angle decides: under 90 degrees is acute, exactly 90 is right, over 90 is obtuse. From sides alone, with c longest: a²⁺b² = c² means right, greater means acute, smaller means obtuse.
The equilateral triangle. For perimeter 30, the equilateral (10, 10, 10) has area about 43.3, while a 5-12-13 right triangle with the same perimeter has area 30, noticeably smaller.
Equilateral: all three sides equal (and all angles 60 degrees). Isosceles: two equal sides with two equal base angles. Scalene: no equal sides and no equal angles. Every triangle fits exactly one of these.
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