Three measurements pin down a triangle, as long as at least one of them is a side. This calculator handles all five combinations that work (SSS, SAS, ASA, AAS and the tricky SSA) and returns the missing sides and angles plus the area, perimeter, triangle type, every height, median and angle bisector, and the inradius and circumradius. When SSA data fits two different triangles, you get both.
How to use the triangle calculator
- In What do you know?, pick the case that matches your data.
- Enter the values that appear. Side a is opposite angle A, b is opposite B and c is opposite C, so label your sketch first.
- Choose the Units for sides and the Angle unit (degrees by default, or radians).
- Read the tape: values you entered are marked “given” and solved values are highlighted. The table underneath lists heights, medians, angle bisectors and both circle radii, and the diagram is drawn to scale.
The five solvable cases
| Case | You enter | How it is solved |
|---|---|---|
| SSS | Sides a, b, c | Law of cosines for A and B, then C = 180° − A − B |
| SAS | Sides a, b and angle C between them | Law of cosines for side c, then the angles |
| ASA | Angles A, B and side c between them | C = 180° − A − B, then the law of sines |
| AAS | Angles A, B and side a, opposite A | C = 180° − A − B, then the law of sines |
| SSA | Sides a, b and angle A, opposite a | Law of sines, giving 0, 1 or 2 triangles |
The area is ½bc sin A, the inradius is r = area ÷ s (where s is half the perimeter), and the circumradius is R = a ÷ (2 sin A).
Why three angles are not enough
Angles fix a triangle’s shape but not its size. A triangle with angles of 40°, 60° and 80° can have a shortest side of 1 cm or 1 km; all of them are similar, scaled copies of one another. Only two of the three angles carry information anyway, since the third is always 180° minus the other two. Add one side and the problem becomes ASA or AAS.
The SSA ambiguous case
Picture angle A at the left end of a base line, side b rising from it, and side a hanging from the top of b like a pendulum. Whether a reaches the base, and how many times, depends on the height h = b·sin A:
- a < h: side a is too short to reach the base, so no triangle exists.
- a = h: side a just touches the base, giving one right triangle.
- h < a < b: side a crosses the base at two points, giving two triangles.
- a ≥ b: the second crossing falls behind angle A, leaving one triangle.
If A is 90° or more, there is one triangle when a > b and none otherwise, because the largest angle must face the longest side.
Worked example: two triangles from one set of data
Choose SSA and enter a = 5 cm, b = 6 cm and A = 40°.
- Height test: h = 6 × sin 40° ≈ 3.857 cm. Because 3.857 < 5 < 6, two triangles fit.
- Law of sines: sin B = 6 × sin 40° ÷ 5 ≈ 0.7713, so B ≈ 50.47° or B ≈ 180° − 50.47° = 129.53°.
- Triangle 1: B ≈ 50.47°, C ≈ 89.53°, c ≈ 7.778 cm, area ≈ 15.00 cm².
- Triangle 2: B ≈ 129.53°, C ≈ 10.47°, c ≈ 1.414 cm, area ≈ 2.727 cm².
Both appear side by side. To choose one, you need one more fact, such as whether B is acute or obtuse. Shorten a to 3 cm and no triangle fits; lengthen it to 7 cm and only one does, with B ≈ 33.43°.
Error messages for impossible inputs
- “These sides can’t form a triangle.” The two shorter sides must add up to more than the longest. Sides 3, 4 and 8 fail because 3 + 4 = 7 is less than 8; sides 3, 4 and 7 fail too, because they lie flat.
- “Angles A and B add up to …” In ASA and AAS the two given angles must total less than 180°, leaving room for C. Angles of 100° and 80° leave nothing.
- “No triangle fits.” In SSA, side a is shorter than b·sin A, or angle A is right or obtuse while a is not longer than b.
- Out-of-range values. Every side must be greater than zero, and every angle greater than 0° and less than 180° (π rad).
Reading the extra measures
Heights (altitudes) are perpendicular distances from each corner to the opposite side. Medians join each corner to the midpoint of the opposite side and meet at the centroid, the balance point. Angle bisectors split each angle in half and meet at the incenter, the center of the inscribed circle. Triangle type pairs the side class (scalene, isosceles, equilateral) with the angle class (acute, right, obtuse).
The isosceles and equilateral triangle calculators need fewer inputs, and the triangle area calculator also works from base and height or corner coordinates.
Frequently asked questions
Why can't I solve a triangle from three angles?
Three angles fix only the shape. Every triangle with angles of 40°, 60° and 80° is a scaled copy of every other one, so the sides could be any size. Add at least one side and use the ASA or AAS option.
What is the SSA ambiguous case?
When you know two sides and an angle that is not between them, the data can fit no triangle, one triangle or two. Compare side a with the height h = b·sin A: shorter than h gives none, between h and b gives two, and exactly h or at least b gives one. The calculator runs this test and shows both triangles when two exist.
Why does the calculator say my sides can't form a triangle?
The two shorter sides must add up to more than the longest side; this is the triangle inequality. Sides of 3, 4 and 8 fail because 3 + 4 = 7 is less than 8. If the sum is exactly equal, the shape collapses into a flat line with no area.
What is the difference between ASA and AAS?
Both use two angles and one side. In ASA the side lies between the two known angles (side c, between A and B); in AAS it faces one of them (side a, opposite A). In both cases the calculator finds the third angle first and then uses the law of sines.
What are the inradius and circumradius?
The inradius r is the radius of the largest circle that fits inside the triangle, touching all three sides; it equals the area divided by half the perimeter. The circumradius R is the radius of the circle through all three corners, R = a ÷ (2 sin A). For a triangle with sides 5, 6 and 7, r ≈ 1.633 and R ≈ 3.572.