Triangle Calculator — Solve Sides, Angles, Area & Perimeter
Three values are all it takes to unlock a whole triangle. Enter any three sides or angles — at least one of them a side — and get every missing side, angle, area, perimeter and radius, drawn to scale.
What you know
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The solved triangle
Your triangle, drawn to scale
The solved triangle at its true proportions, with every side and angle labelled.
- Three values are all it takes — any mix of sides and angles, as long as at least one of them is a side — to unlock every missing side, angle, area, perimeter and radius, drawn to scale.
- The law of cosines cracks side-side-side and side-angle-side; the law of sines handles the rest — and area comes from Heron’s formula, which needs only the three sides.
- Give the solver two sides and an angle that is not between them (SSA) and the numbers can describe two different triangles — the solver flags the ambiguous case when both fit.
Keep exploring
Three companion tools for shapes and lengths.
How it works
Three values in, the whole triangle out.
Enter three values
Any mix of sides and angles — three sides, two sides and the angle between them, one side and two angles — as long as at least one value is a side.
The solver picks the law
The law of cosines cracks side-side-side and side-angle-side; the law of sines handles the rest — and flags the SSA case when two different triangles fit.
Read the full picture
All sides and angles, area, perimeter, the inradius and circumradius, a plain-English type label — and a drawing at true scale.
The math behind the solver
Two laws do all the work. The law of cosines generalises Pythagoras: it finds a third side from two sides and the angle between them, and rearranged, it finds any angle from three sides. The law of sines says every side is proportional to the sine of the angle opposite it, so one known side-angle pair unlocks the rest. Area comes from Heron’s formula, which needs only the three sides.
Worked examples:
| Given | Key step | Result |
|---|---|---|
| Sides 3, 4, 5 (SSS) | cos A = (16 + 25 − 9) ÷ 40 = 0.8 → A = 36.87° · Heron: s = 6, √(6 × 3 × 2 × 1) | Angles 36.87°, 53.13°, 90.00° · Area 6.00 |
| b = 7, c = 5, A = 60° (SAS) | a² = 49 + 25 − 2 × 7 × 5 × cos 60° = 39 · Area = ½ × 7 × 5 × sin 60° | a = 6.2450 · Area 15.16 |
| A = 45°, B = 60°, c = 10 (ASA) | C = 180 − 45 − 60 = 75° · a = 10 × sin 45° ÷ sin 75° | a = 7.3205, b = 8.9658 · Area 31.70 |
The two radii: the inradius r = Area ÷ s is the radius of the largest circle that fits inside the triangle; the circumradius R = abc ÷ 4 × Area is the radius of the circle passing through all three corners. For the 3-4-5 triangle, r = 6 ÷ 6 = 1.00 and R = 60 ÷ 24 = 2.50 — exactly half the hypotenuse, as it always is for a right-angled triangle.
Medians, altitudes and the semiperimeter: the semiperimeter s is half the perimeter — the same s that Heron’s formula and the inradius use, and it is now shown in the results with the other measures. A median runs from one corner to the midpoint of the opposite side; the median to side a is ½√(2b² + 2c² − a²), with the letters swapped for the other two medians. An altitude is the perpendicular height from a corner onto the opposite side: h = 2 × Area ÷ base. For the 3-4-5 triangle the medians are 4.27, 3.61 and 2.50, and the altitudes are 4.00, 3.00 and 2.40.
Go deeper
The famous ambiguous case — and the two circles hiding in every triangle.
Give the solver two sides and an angle that is not between them (SSA) and the numbers can describe two different triangles — or none:
- Why it happens: the law of sines gives sin B, and two angles share the same sine — B and 180° − B. Both can be genuine.
- A concrete case: A = 30°, a = 5, b = 8. Then sin B = 8 × sin 30° ÷ 5 = 0.8, so B = 53.13° (giving C = 96.87°, c = 9.9282) or B = 126.87° (giving C = 23.13°, c = 3.9282). Two real triangles, one set of inputs.
- When there is only one: if the side opposite the given angle is the longer of the two sides, the obtuse option breaks the 180° budget and only the acute triangle survives.
- When there are none: if sin B comes out above 1, the side is too short to reach — the solver tells you instead of inventing a triangle.
Every triangle hides exactly two special circles, and this calculator reports both radii:
- The incircle is the largest circle that fits inside, touching all three sides. Its radius is r = Area ÷ s, where s is half the perimeter.
- The circumcircle passes through all three vertices. Its radius is R = abc ÷ 4 × Area — and by the law of sines, R is also a ÷ (2 sin A) for any corner.
- A right-triangle shortcut: the hypotenuse is a diameter of the circumcircle, so R is always half the hypotenuse — 2.50 for sides 3, 4, 5.
- Where they show up: incircles size the largest round object that fits a triangular plot or part; circumcircles appear in surveying, gear design and GPS trilateration.
Choosing the right three values: SSS, SAS, ASA, SSA, and why AAA fails
Three values solve a triangle — but only the right combinations, and one combination never does.
With three sides (SSS), the triangle is fixed up to congruence: the law of cosines finds the angles, and the area follows from Heron’s formula. With two sides and the angle between them (SAS), the third side comes from the law of cosines and the rest follows. With one side and two angles (ASA or AAS), the third angle is 180° minus the other two, and the law of sines finds the remaining sides. With two sides and a non-included angle (SSA), the answer may be one triangle, two triangles, or none — the ambiguous case — because the sine of an angle equals the sine of its supplement, so the same numbers can build two different shapes.
Three angles (AAA) never solve a triangle, because angles fix the shape but not the size: a triangle with angles 45°, 60° and 75° can be the size of a coin or the size of a field. At least one side must be known to set the scale. The solver on this page therefore needs exactly three values including at least one side, and it reports when the values are inconsistent — sides that violate the triangle inequality, or angles that already reach 180° — rather than inventing a triangle that cannot exist.
Area three ways: base-height, Heron, and two sides with an included angle
The area formula you need depends on the values you have. All three give the same answer on the same triangle.
With a base and its height, Area = ½ × base × height. With three sides, Heron’s formula uses the semiperimeter s = (a + b + c) ÷ 2: Area = √(s(s − a)(s − b)(s − c)). For sides 3, 4 and 5, s = 6 and the area is √(6 × 3 × 2 × 1) = √36 = 6. With two sides and the angle between them, Area = ½ × a × b × sin(C). The three formulas agree wherever they overlap, which makes them a useful cross-check: if base-height and Heron disagree, one of the inputs is wrong, not the geometry.
The inradius and circumradius complete the picture. The inradius — the radius of the largest circle that fits inside the triangle — is r = Area ÷ s. The circumradius — the radius of the circle through all three corners — is R = abc ÷ (4 × Area). Both are shown by the solver, and both scale with the triangle: double every side and both radii double, while the area quadruples. That scaling behaviour is itself a check on any answer that looks too large or too small.
Frequently asked questions
Quick answers about solving triangles.
How many values do I need to solve a triangle?
Exactly three, and at least one of them must be a side. Three sides (SSS), two sides with the angle between them (SAS), one side with two angles (ASA or AAS) and two sides with an angle opposite one of them (SSA) all work. This calculator accepts the three values in any of the six boxes and leaves the rest to the solver.
What is the ambiguous case in SSA?
When you give two sides and an angle that is not between them (SSA), the same numbers can sometimes build two different triangles, because sin B and sin (180° − B) are equal. Example: A = 30°, a = 5, b = 8 gives B = 53.13° or B = 126.87°, both valid. When that happens here, the first solution is shown and the second is spelled out in the note under the results.
How do you find the area of a triangle without the height?
With Heron’s formula, which needs only the three sides. Halve the perimeter to get s, then Area = √(s(s − a)(s − b)(s − c)). For sides 3, 4 and 5: s = 6, so Area = √(6 × 3 × 2 × 1) = √36 = 6.00. If you know two sides and the angle between them instead, Area = ½ × b × c × sin A works too.
What is the difference between the inradius and the circumradius?
The inradius is the radius of the largest circle that fits inside the triangle, touching all three sides: r = Area ÷ s. The circumradius is the radius of the circle that passes through all three corners: R = abc ÷ (4 × Area). In a right-angled triangle the circumradius is simply half the hypotenuse — 2.50 in the 3-4-5 triangle.
Can three angles alone define a triangle?
They define its shape but not its size. Angles of 45°, 60° and 75° describe a whole family of similar triangles, from one the size of a coin to one the size of a field — the angles fix the proportions, but with no side length there is nothing to set the scale. That is why this calculator always asks for at least one side among your three values.
Why does the calculator say my triangle is impossible?
Usually one of three reasons. The triangle inequality: the two shorter sides must add up to more than the longest side, so sides of 2, 3 and 6 can never meet. The angle budget: the three angles must total exactly 180°, so two given angles of 100° and 90° leave nothing for the third. Or an SSA dead end: the side opposite your angle is too short to reach the other side, making sin B larger than 1. The error message names which rule your numbers broke.
What is the 30-60-90 shortcut?
In a right triangle with angles 30°, 60° and 90°, the sides are always in the ratio 1 : √3 : 2 — shortest side opposite 30°, longer leg opposite 60°, hypotenuse twice the shortest side. If the shortest side is 5, the other leg is 5√3 (about 8.66) and the hypotenuse is 10, with no trigonometry needed. The 45-45-90 triangle has the parallel shortcut 1 : 1 : √2. Both shortcuts are consequences of the Pythagorean theorem applied to half an equilateral triangle and half a square, and both are exact, not approximations.
How do I check a solved triangle?
Three checks, in order. First, the angles must add to 180° — any other total means an error, not a special triangle. Second, the sides must satisfy the triangle inequality: the two shorter sides together must exceed the longest side, or the triangle cannot close. Third, the law of sines must hold consistently: a ÷ sin(A) = b ÷ sin(B) = c ÷ sin(C), so each side divided by the sine of its opposite angle gives the same number. If any check fails, the inputs are inconsistent and the triangle as described does not exist.
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Disclaimer
This calculator is for information and education only. Results are computed with standard trigonometry and rounded for display — angles can be entered and read in degrees or radians; for engineering, construction or surveying work, verify critical measurements with professional tools and allow for real-world tolerances.