This free online triangle calculator solves any triangle from the information you already have: three sides (SSS), two sides and the included angle (SAS), one side and two angles (ASA / AAS), or the two legs of a right triangle. Every mode returns the remaining sides and angles plus the area, perimeter, semiperimeter, inradius, circumradius, height and the triangle type — with the formulas shown below and a to-scale diagram you can drag.
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Right TriangleCalculation Details
Detailed Parameters
| Side a | 30 |
|---|---|
| Side b | 40 |
| Side c | 50 |
| Semiperimeter | 60 |
| Inradius | 10 |
| Circumradius | 25 |
Internal Angles
What Is a Triangle and How Is It Labelled?
A triangle is a closed three-sided polygon whose three interior angles always add up to 180°. The calculator uses the standard convention: sides are named a, b and c, and each angle is named with the same capital letter as the side it faces, so a faces A, b faces B and c faces C. In a right triangle the side facing the 90° angle is the hypotenuse.
Triangles are classified by their angles and by their sides, and the tool tells you which type you have as soon as it has the numbers. If one angle is 90° it reports a Right Triangle; if all three sides are equal it is an Equilateral Triangle; if two sides are equal it is an Isosceles Triangle; if every angle is under 90° it is an Acute Triangle; otherwise it is an Obtuse Triangle. That single label is why the same page works as an equilateral triangle calculator, an isosceles triangle angle calculator and a right triangle calculator without switching tools.
Triangle Formulas and Principles
Everything the tool does comes from five classical identities. Each mode picks the ones it needs, and the tool evaluates them directly without iterative guessing.
| Symbol | Meaning | How it is obtained |
|---|---|---|
| a, b, c | the three side lengths | entered, or solved from the other five identities |
| A, B, C | the angles opposite a, b, c | A + B + C = 180° |
| s | semiperimeter | s = (a + b + c) / 2 |
| S | area | Heron’s formula, or ½·base·height, or ½·ab·sin C |
| h | height | h = 2S / a (from side a; from the hypotenuse in the right-triangle mode) |
| r | inradius (incircle) | r = S / s |
| R | circumradius (circumcircle) | R = abc / (4S), or R = a / (2 sin A) |
Angle sum. The three interior angles always close the triangle:
\[ A + B + C = 180^\circ \]
Heron’s formula (area from three sides). The area of triangle with 3 sides is exactly what Heron’s formula solves, and it is the triangle area formula the tool uses in the SSS mode. Work out the semiperimeter first, then:
\[ s = \frac{a+b+c}{2}, \qquad S = \sqrt{s(s-a)(s-b)(s-c)} \]
Law of cosines (angles from three sides). This is how the tool recovers every angle once all three sides are known:
\[ \cos A = \frac{b^2 + c^2 – a^2}{2bc}, \qquad \cos B = \frac{a^2 + c^2 – b^2}{2ac}, \qquad \cos C = \frac{a^2 + b^2 – c^2}{2ab} \]
Law of sines. When only one side and two angles are known, the other two sides follow from the sine rule — and the same ratio gives the circumradius:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R \]
Two sides and the included angle (SAS). The third side comes from the cosine rule, and the area from the sine of the angle between the two known sides:
\[ c = \sqrt{a^2 + b^2 – 2ab\cos C}, \qquad S = \frac{1}{2}ab\sin C = \frac{bc\sin A}{2} = \frac{ca\sin B}{2} \]
Right triangles. When C = 90°, the cosine rule collapses into the Pythagorean theorem, which is the identity behind every pythagorean theorem calculator:
\[ c = \sqrt{a^2 + b^2}, \qquad a^2 + b^2 = c^2, \qquad S = \frac{ab}{2}, \qquad R = \frac{c}{2} \]
The complete formula for triangle work therefore always has the same shape: find one missing quantity with the identity that fits the data you have, then propagate it. A hypotenuse calculator only ever needs the two legs, and because the mode recovers everything else from those two numbers, the same page is a right triangle finder and a right triangle solver as well: feet or metres in, all three angles and the area out.
Special right triangles. Two families come up constantly in tests and in the field, and both are pure ratios, so you can use this page as a 45 45 90 triangle calculator or a 30 60 90 triangle calculator. A 45 45 90 triangle has equal legs and a hypotenuse of leg × √2, giving the ratio 1 : 1 : 1.414; a 30 60 90 triangle has sides in the ratio 1 : 1.732 : 2, with the shortest side always opposite the 30° angle. Verify either one by entering the two legs, or one side with the two angles.
| Special triangle | Ratio of sides | Angles | Typical use |
|---|---|---|---|
| 45 45 90 | 1 : 1 : √2 | 45°, 45°, 90° | square diagonals, roof pitch, CAD layouts |
| 30 60 90 | 1 : √3 : 2 | 30°, 60°, 90° | equilateral halves, truss and ramp geometry |
| 3 4 5 | 3 : 4 : 5 | 36.87°, 53.13°, 90° | squaring up corners, setting out on site |
Input and output mapping. Input: three sides; or two sides plus the angle between them; or one side plus the two angles at its ends; or the two legs of a right triangle. Output in every mode: the other sides, all three angles, the area, perimeter, semiperimeter, inradius, circumradius, height and the triangle type.
Precision and boundaries. Every length must be greater than zero, and every angle must be between 0° and 180°. In the three-sides mode the triangle inequality must hold — the sum of any two sides has to be greater than the third — and in the angles mode the two angles you enter must add up to less than 180°, otherwise no triangle exists and the tool rejects the input. Results are rounded to two decimals, whole numbers are displayed without a decimal point (a semiperimeter of exactly 6 stays 6 rather than 6.00), and angles are shown in degrees. The calculation is unitless: enter every length in the same unit — millimetres, centimetres, inches or feet — and the results come back in that unit, with the area in square units. Note that the reported height is measured from side a in the SSS, SAS and ASA modes, and from the hypotenuse in the right-triangle mode — the Details card names the side it belongs to.
How to Use the Triangle Calculator
- Three sides (SSS) — enter a, b and c. Use this when you can measure all three sides, or when you want to check whether a triangle with those sides is valid at all; this is the SSS triangle calculator path when the angles are unknown.
- Two sides and the included angle (SAS) — enter a, b and the angle C between them. This is the mode for field measurements, where the two sides are easy to measure but the third is not.
- One side and two angles (ASA / AAS) — enter side a and the angles B and C; angle A is filled in automatically as 180° − B − C.
- Right triangle — enter the two legs a and b; the hypotenuse is shown as it is computed.
- Choose the mode that matches your data in the Calculation Mode dropdown. Only the input fields for that mode stay on screen.
- Enter the numbers: type them, or drag the slider under a field — the two are linked, so either one recalculates the triangle immediately.
- Press Calculate to refresh the results explicitly; the Calculation Details card shows area, perimeter and height, the Detailed Parameters block shows the semiperimeter, inradius and circumradius, and the Internal Angles block lists A, B and C.
- Check the diagram: it is drawn to scale with the corners labelled, and you can drag to rotate it, zoom in and out, or reset the view.
- Keep the result with Share Results — the link carries the values you entered, so reopening it restores the same triangle — or generate a PNG of the diagram and the full result set. Reset returns the form to its defaults.
Worked Examples
The six examples below cover all four modes. Every number is what the calculator itself displays for that input, rounded to two decimals.
Example 1 — three sides (3, 4, 5): the classic right triangle
With a = 3, b = 4 and c = 5 the triangle inequality holds, so the semiperimeter is s = (3 + 4 + 5) / 2 = 6 and Heron’s formula gives S = √(6 × 3 × 2 × 1) = 6. The law of cosines returns the angles: A = 36.87°, B = 53.13° and C = 90°, so the tool labels it a Right Triangle with an inradius of 1 and a circumradius of 2.5 — the circumradius is exactly half the hypotenuse, as it must be for any right triangle.
| Quantity | Value |
|---|---|
| Sides a, b, c | 3, 4, 5 |
| Angles A, B, C | 36.87°, 53.13°, 90° |
| Area | 6 |
| Perimeter | 12 |
| Semiperimeter | 6 |
| Inradius | 1 |
| Circumradius | 2.5 |
| Height (from side a) | 4 |
| Type | Right Triangle |
Example 2 — two sides and the included angle (7, 9, 45°): find the missing side of a triangle
This mode answers the everyday question “how do you find the missing side of a triangle” when you can measure two sides and the corner between them. With a = 7, b = 9 and C = 45°, the cosine rule gives c = √(49 + 81 − 2 × 7 × 9 × cos 45°) = 6.4. The area is ½ × 7 × 9 × sin 45° = 22.27, the remaining angles are A = 50.71° and B = 84.29°, and the perimeter is 22.4 with a semiperimeter of 11.2.
| Quantity | Value |
|---|---|
| Side c | 6.4 |
| Angles A, B, C | 50.71°, 84.29°, 45° |
| Area | 22.27 |
| Perimeter | 22.4 |
| Semiperimeter | 11.2 |
| Inradius | 1.99 |
| Circumradius | 4.52 |
| Height (from side a) | 6.36 |
| Type | Acute Triangle |
Example 3 — one side and two angles (10, 30°, 100°): an obtuse triangle
With a = 10, B = 30° and C = 100°, the third angle is A = 180° − 30° − 100° = 50°, and the law of sines gives b = 10 × sin 30° / sin 50° = 6.53 and c = 10 × sin 100° / sin 50° = 12.86. The area is 32.14, the perimeter 29.38 and the circumradius 6.53 — which equals side b here, because b faces the 50° angle and a / sin A is the same ratio for every side. Because one angle is larger than 90°, the tool labels it an Obtuse Triangle.
| Quantity | Value |
|---|---|
| Sides a, b, c | 10, 6.53, 12.86 |
| Angles A, B, C | 50°, 30°, 100° |
| Area | 32.14 |
| Perimeter | 29.38 |
| Semiperimeter | 14.69 |
| Inradius | 2.19 |
| Circumradius | 6.53 |
| Height (from side a) | 6.43 |
| Type | Obtuse Triangle |
Example 4 — right triangle from two legs (5 and 12): hypotenuse and altitude
Enter the legs a = 5 and b = 12 and the tool acts as a hypotenuse calculator: c = √(25 + 144) = 13. The area is ½ × 5 × 12 = 30, the perimeter is 30 as well, the semiperimeter is 15, the inradius is 2 and the circumradius is 6.5. The two acute angles are A = 67.38° and B = 22.62°, and the reported height is the altitude to the hypotenuse, 2 × 30 / 13 = 4.62 — not one of the legs, which is the one place this mode differs from the SSS mode.
| Quantity | Value |
|---|---|
| Sides a, b, c | 5, 12, 13 |
| Angles A, B, C | 67.38°, 22.62°, 90° |
| Area | 30 |
| Perimeter | 30 |
| Semiperimeter | 15 |
| Inradius | 2 |
| Circumradius | 6.5 |
| Height (from the hypotenuse) | 4.62 |
| Type | Right Triangle |
Example 5 — 45 45 90 triangle with legs of 1
Equal legs of 1 give the smallest of the special right triangles: the hypotenuse is 1.41, the area is 0.5, the perimeter is 3.41 and the semiperimeter is 1.71. Both acute angles come out at exactly 45°, the inradius is 0.29 and the circumradius is 0.71. Scale every number by the leg length and you have the whole 45 45 90 family: legs of 10 give a hypotenuse of 14.14 and an area of 50.
| Quantity | Value |
|---|---|
| Sides a, b, c | 1, 1, 1.41 |
| Angles A, B, C | 45°, 45°, 90° |
| Area | 0.5 |
| Perimeter | 3.41 |
| Semiperimeter | 1.71 |
| Inradius | 0.29 |
| Circumradius | 0.71 |
| Height (from the hypotenuse) | 0.71 |
| Type | Right Triangle |
Example 6 — 30 60 90 triangle from one side and two angles
Enter a = 2 with B = 60° and C = 90°: the remaining angle is A = 30°, and the sine rule gives b = 3.46 and c = 4. That is the 1 : 1.732 : 2 ratio in action — the shortest side is 2, opposite the 30° angle, and the hypotenuse is exactly double it. The area is 3.46, the perimeter 9.46 and the inradius 0.73.
| Quantity | Value |
|---|---|
| Sides a, b, c | 2, 3.46, 4 |
| Angles A, B, C | 30°, 60°, 90° |
| Area | 3.46 |
| Perimeter | 9.46 |
| Semiperimeter | 4.73 |
| Inradius | 0.73 |
| Circumradius | 2 |
| Height (from side a) | 3.46 |
| Type | Right Triangle |
Triangle Calculator FAQ
How do I find the area of a triangle?
Use the triangle area formula that matches your data. With a base and a height, S = ½ × base × height. With all three sides, use Heron’s formula through the semiperimeter. With two sides and the angle between them, S = ½ × a × b × sin C. The find area of triangle formula you need is therefore never more than one identity away, and this calculator applies the right one automatically for whichever mode you pick.
What is Heron’s formula and when do I need it?
Heron’s formula is S = √(s(s − a)(s − b)(s − c)) with s = (a + b + c) / 2, and it is the only classical formula that turns three side lengths into an area without any angle. You need it whenever measuring an angle is impractical — a plot of land, a sail, a bent sheet-metal part — and it is what the three-sides mode runs. The 3-4-5 triangle above is the smallest integer case: s = 6 and S = 6.
How do I find the missing side of a triangle?
It depends on what else you know. If you have two sides and the angle between them, use the cosine rule — that is Example 2 above, where c came out at 6.4 from a = 7, b = 9 and C = 45°. If you have one side and two angles, use the sine rule. If it is a right triangle, the Pythagorean theorem settles it in one step. As a triangle side finder, the calculator covers all three cases without you having to remember which rule applies.
How do I find a missing angle?
With all three sides, take the inverse cosine of the law of cosines for each corner; with two angles, subtract their sum from 180°. An angle finder for triangle work only needs one of those two situations, and the tool shows all three angles at once — for the 3-4-5 triangle it returns 36.87°, 53.13° and 90° without you selecting an angle first. The Internal Angles block is also the quickest check that your input was consistent: the three values always add up to 180°.
What is the hypotenuse and how do I calculate it?
The hypotenuse is the longest side of a right triangle, opposite the 90° angle, and it equals √(a² + b²) for legs a and b. Legs of 5 and 12 give 13; legs of 1 and 1 give 1.41. As a hypotenuse calculator this page needs only the two legs, and as a right triangle solver it also returns the two acute angles, the area, the inradius and the circumradius — for any right triangle the circumradius is exactly half the hypotenuse, which is a useful sanity check on the numbers.
Can I solve for x on triangle problems with only two values?
Almost always, yes — as long as one of the two is a side. Two sides with the angle between them, one side with two angles, or the two legs of a right triangle each determine the shape completely, so there is exactly one triangle and one answer for x. Two angles alone do not: they fix the shape but not the size, so the sides can be scaled freely. That is why every mode here asks for at least one length.
How do I calculate square feet of a triangle?
Measure a base and the perpendicular height in feet and halve the product: a triangular bed with a 120 ft base and an 80 ft height has an area of ½ × 120 × 80 = 4800 ft². If you cannot reach the height, measure the three sides instead and let Heron’s formula do it. To turn that into acres, divide by 43,560 — 4800 ft² is 0.11 acre. As a triangle acreage calculator the workflow is the same, only the unit changes.
Does the calculator give a triangle volume?
No — a triangle is a flat shape, so it has an area and no volume, and any “triangle volume formula” you see refers to a solid. The solid you probably want is a triangular prism (or a pyramid), and both start from the triangle area printed above: a prism’s volume is area × length, so the 3-4-5 triangle extruded 10 units deep encloses 60 cubic units, while a pyramid over the same base and height divides by three. Calculate the flat triangle here first, then apply the solid’s own factor.
Why does the tool say my triangle is invalid?
Two inputs are impossible by construction, and the calculator refuses both rather than returning a wrong answer. In the three-sides mode the triangle inequality must hold: if any single side is as long as or longer than the sum of the other two, the three lengths cannot meet, so 1, 2, 10 is rejected while 3, 4, 5 is accepted. In the angles mode the two angles must add up to less than 180°, otherwise the third angle would be zero or negative and there would be no triangle at all.
Related Tools
This calculator belongs to the Math cluster on BunTool, where every page follows the same structure: definition, formulas, worked examples, FAQ. If your shape is a regular polygon rather than a triangle, the pentagon calculator and the hexagon calculator solve those from a single known value using the same formulas-and-examples layout. The full set of geometry and algebra tools lives under Math.
