Triangle Calculator

Solve triangle sides, angles, and area from any valid set of three known values, including the ambiguous SSA case.

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Select what you know about the triangle, then enter the values to calculate the rest.

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Calculation Examples

Calculation Case Result
Sides: 3, 4, 5 (Right Triangle) Area = 6
Equilateral triangle, side 6 Area = 15.588
Hypotenuse from legs 5 and 12 13
SSA: a = 7, b = 10, A = 40° Two triangles: B = 66.7°, C = 73.3°, c = 10.43 or B = 113.3°, C = 26.7°, c = 4.89
Mihail Doncenco
Mihail Doncenco Site Creator & Calculator Expert

Founder and developer of CalcMate. Builds and maintains every calculator on this site, focusing on formula accuracy and clear, practical results.

How to Solve a Triangle?

To solve a triangle, enter at least three known values in any valid combination: SSS, SAS, ASA, AAS, or SSA. At least one of them must be a side, because three angles fix the shape but not the size. The tool applies the Pythagorean theorem, \(a^2 + b^2 = c^2\), to right triangles and the Law of Sines and Law of Cosines to oblique ones. For the ambiguous SSA case, it shows both solutions when two valid triangles exist.Solving triangles with AAS and ASA formulas

How This Calculator Works

The calculator finds all unknown sides and angles, plus the area, perimeter, altitudes, and medians. SSS uses the Law of Cosines to get each angle. ASA and AAS use the 180° angle sum and the Law of Sines. Right triangles combine the Pythagorean theorem with trigonometric ratios. These laws are classical: the Pythagorean theorem is Proposition 47 in Book I of Euclid Elements, the Law of Cosines generalizes Propositions 12 and 13 in Book II, and Heron of Alexandria gave his area formula in the Metrica (1st century CE).

SSA needs the most care. With sides \(a\) and \(b\) and the angle \(A\) opposite \(a\), the outcome depends on \(b \sin A\): if \(a < b \sin A\) there is no triangle, if \(a = b \sin A\) there is exactly one (a right triangle), if \(b \sin A < a < b\) there are two, and if \(a \geq b\) there is one. Example: \(a = 7\), \(b = 10\), \(A = 40^\circ\) gives \(b \sin A = 6.43\), which is less than 7 and less than 10, so two triangles exist. One has \(B = 66.7^\circ\), \(C = 73.3^\circ\), \(c = 10.43\), and the other has \(B = 113.3^\circ\), \(C = 26.7^\circ\), \(c = 4.89\).

Right angled triangle diagram

Useful Tips 💡

  • Check that the three angles add up to 180° and that any two sides add up to more than the third. These two checks catch most input errors.
  • For SSA inputs, compare both solutions before you choose. With a = 7, b = 10, and A = 40°, side c is either 10.43 or 4.89, and only a drawing or a field measurement tells you which triangle you have.
  • When you test for a right triangle, treat the longest side as c. It must satisfy a² + b² = c², for example 5² + 12² = 13².

📋Steps to Calculate

  1. Enter at least three known values (sides or angles, or a combination).

  2. Select triangle type if known (right, oblique, equilateral).

  3. Click "Calculate" to receive all missing sides, angles, area, and perimeter.

Mistakes to Avoid ⚠️

  1. Entering sides that break the triangle inequality, such as 1, 2, and 10. Since 1 + 2 = 3 is less than 10, the two short sides cannot reach each other to close the triangle.
  2. Assuming SSA always gives one triangle. Depending on the numbers, it can give none, one, or two, so always check for a second solution.
  3. Entering angles that do not add up to 180°, such as 60°, 70°, and 60° (a total of 190°). Also remember that three angles alone give the shape of a triangle, not its size, so add at least one side.
  4. Using a² + b² = c² on a triangle without a right angle. The theorem holds only when the angle opposite c is 90°. For any other triangle, use the Law of Cosines.

Practical Applications📊

  1. Right triangle area: a 3-4-5 triangle, a classic Pythagorean triple, has legs of 3 and 4, so its area is 3 × 4 ÷ 2 = 6 square units.

  2. Hypotenuse from two legs: legs of 5 and 12 give a hypotenuse of √(25 + 144) = 13, another Pythagorean triple that is handy for quick checks.

  3. Equilateral triangle area: a side of 6 gives (√3 ÷ 4) × 36 ≈ 15.59 square units.

  4. Surveying with ASA: from a 100 m baseline with angles of 70° and 50° at its two ends, the third angle is 60°. The Law of Sines gives the other sides as 108.5 m and 88.5 m, and the area is about 4,156 m², all without measuring those sides directly.

Questions and Answers

Can the Triangle Calculator support 3D geometry problems like tetrahedrons?

No, the calculator solves triangles in a flat 2D plane, but its results feed directly into 3D problems. The area of a triangular face, a base edge, or the height of a cross-section are standard inputs for volume formulas. For a regular tetrahedron with edge \(a\), the volume is \(V = \dfrac{1}{3} \times \text{base area} \times \text{height} = \dfrac{a^3}{6\sqrt{2}}\), where the base is an equilateral triangle with area \(\dfrac{\sqrt{3}}{4}a^2\). For full 3D solids, use a dedicated solid geometry tool alongside these 2D results.

How do I calculate the area of a triangle using different methods?

The right area formula depends on what you know. With base \(b\) and perpendicular height \(h\), use \(A = \dfrac{1}{2}bh\). With three sides and no height, use Heron formula: first the semi-perimeter \(s = \dfrac{a+b+c}{2}\), then \(A = \sqrt{s(s-a)(s-b)(s-c)}\). With two sides and the included angle (SAS), use \(A = \dfrac{1}{2}ab\sin C\). Example: sides 13, 14, 15 give \(s = 21\) and \(A = \sqrt{21 \times 8 \times 7 \times 6} = 84\). For \(a = 8\), \(b = 5\), \(C = 60^\circ\), \(A = \dfrac{1}{2} \times 8 \times 5 \times \sin 60^\circ = 17.32\).

How do I find triangle sides from angles and one known side?

Enter the known side and two angles (ASA or AAS), and the calculator applies the Law of Sines. If side \(a\) and angles \(A\) and \(B\) are known, then \(C = 180^\circ - A - B\), \(b = \dfrac{a \sin B}{\sin A}\), and \(c = \dfrac{a \sin C}{\sin A}\). Example: a 100 m baseline with angles \(70^\circ\) and \(50^\circ\) at its ends has a third angle of \(60^\circ\), and the other sides are \(\dfrac{100 \sin 70^\circ}{\sin 60^\circ} = 108.5\) m and \(\dfrac{100 \sin 50^\circ}{\sin 60^\circ} = 88.5\) m. Surveyors use this to find distances they cannot measure directly.

How do you find a missing angle of a triangle?

If two angles are known, subtract them from 180 degrees: \(C = 180^\circ - A - B\), because the angles of a flat (Euclidean) triangle always add up to 180 degrees. With three sides (SSS), the Law of Cosines gives any angle: \(\cos A = \dfrac{b^2 + c^2 - a^2}{2bc}\). With a side, its opposite angle, and another side (SSA or AAS), the Law of Sines \(\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}\) solves the rest. Example: sides 3, 4, 5 give \(\cos A = \dfrac{16 + 25 - 9}{2 \times 4 \times 5} = 0.8\), so \(A = 36.87^\circ\).

How does a right triangle calculator handle trigonometry?

A right triangle has one 90 degree angle, which links its sides through the Pythagorean theorem \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse. The calculator then finds the two acute angles with inverse trigonometric functions, using \(\sin A = \dfrac{a}{c}\), \(\cos A = \dfrac{b}{c}\), and \(\tan A = \dfrac{a}{b}\), where \(a\) is opposite \(A\) and \(b\) is adjacent. Example: for the 3-4-5 triangle, \(A = \arctan\left(\dfrac{3}{4}\right) = 36.87^\circ\) and \(B = 53.13^\circ\). This is the same math behind navigation bearings, roof pitch, and vector components in physics.

What formulas ensure the mathematical accuracy of this tool?

The calculator relies on three classical results. The Law of Sines, \(\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}\), links each side to the sine of its opposite angle. The Law of Cosines, \(c^2 = a^2 + b^2 - 2ab\cos C\), generalizes the Pythagorean theorem to any triangle (Euclid, Elements, Book II, Propositions 12 and 13). Heron formula, \(A = \sqrt{s(s-a)(s-b)(s-c)}\), gives area from three sides (Heron of Alexandria, Metrica). Altitudes and medians follow from \(h_a = \dfrac{2A}{a}\) and \(m_a = \dfrac{1}{2}\sqrt{2b^2 + 2c^2 - a^2}\).

What is a triangle calculator and what can it solve?

A triangle calculator finds every unknown property of a triangle from three known values, as long as at least one is a side. Given any valid combination of sides and angles (SSS, SAS, ASA, AAS, or SSA), it returns the remaining sides, all three angles, the area, the perimeter, the altitudes, and the medians. It uses the Pythagorean theorem for right triangles and the Law of Sines and Law of Cosines for oblique triangles. Example: sides 13, 14, and 15 give an area of 84 square units.
Disclaimer: This calculator is designed to provide helpful estimates for informational purposes. While we strive for accuracy, financial (or medical) results can vary based on local laws and individual circumstances. We recommend consulting with a professional advisor for critical decisions.