Triangle Calculator

How to Use This Triangle Calculator

Choose Auto or a named case tab. Enter known sides and angles in degrees. Leave unknowns blank. Results show all six parts, area, perimeter, radii, heights, a scaled diagram, and step by step notes.

Auto mode detects SSS, SAS, ASA, AAS, or SSA from whichever three fields you fill. Named tabs only remind you which pattern to enter. They do not block other inputs.

What is a triangle?

A triangle is a three sided polygon. Interior angles always add to 180°. The triangle inequality says any two sides sum to more than the third side. Classifications by sides are equilateral, isosceles, or scalene. Classifications by angles are acute, right, or obtuse.

How to solve a triangle manually

1. Identify the case from the known parts. 2. For SSS use the law of cosines to find each angle. 3. For SAS find the third side with cosines, then another angle with cosines or sines. 4. For ASA or AAS find the third angle first, then use sines for sides. 5. For SSA check the height and the ambiguous conditions before accepting one or two solutions.

Worked SSS example: sides 3, 4, 5. Cosine of C opposite 5: cos C = (3²+4²−5²)/(2·3·4) = 0 → C = 90°. Then A = arccos((4²+5²−3²)/(2·4·5)) ≈ 36.87°, and B ≈ 53.13°. Area = 6.

Worked SAS example

Given a = 8, c = 10, and included angle B = 60°. First find side b with the law of cosines: b² = 8² + 10² − 2·8·10·cos(60°) = 64 + 100 − 160·0.5 = 84, so b ≈ 9.165. Next find angle A with cosines or sines, then C = 180° − A − B. Area = (1/2)·a·c·sin(B) = 0.5·8·10·(√3/2) ≈ 34.64.

Worked ASA and AAS examples

ASA sample: A = 40°, side c = 12 (between A and B), B = 65°. Then C = 75°. Law of sines: a = 12·sin(40°)/sin(75°) ≈ 7.95 and b = 12·sin(65°)/sin(75°) ≈ 11.25. AAS is similar once the third angle is known. The difference is only which side you measured relative to the two angles.

SSA ambiguous case walkthrough

Given a = 7, A = 40°, b = 10. Height from the opposite vertex to side b is h = b·sin(A) ≈ 6.43. Because h < a < b and A is acute, two triangles exist. One uses B = arcsin((b sin A)/a) ≈ 66.82°. The other uses B′ = 180° − 66.82° ≈ 113.18°. Each B produces a different C and side c. The calculator lists both so you can match the sketch your problem intends.

Heron formula for area

When all three sides are known, semiperimeter s = (a+b+c)/2 and Area = √[s(s−a)(s−b)(s−c)]. For 6, 8, 10: s = 12 and Area = √[12·6·4·2] = √576 = 24. Heron is ideal when you have SSS data and do not yet need an angle.

Heights and related lengths

Altitude to side a is ha = 2·Area/a. The same pattern gives hb and hc. These heights appear in Area = (1/2)·base·height and in many design layouts. Median length formulas and angle bisector lengths are related topics, but this tool focuses on the six primary parts plus area based derived values.

Inradius and circumradius

Inradius r = Area / s. Circumradius R = abc / (4·Area). For a right triangle, R equals half the hypotenuse. For 3-4-5, r = 6/6 = 1 and R = 2.5. Equilateral side length s has r = s√3/6 and R = s√3/3, so R = 2r.

Classification by sides and angles

Equilateral means all sides equal and all angles 60°. Isosceles means at least two sides equal, which forces two base angles equal. Scalene means all sides different. Acute means every angle under 90°. Right means one angle is exactly 90°. Obtuse means one angle exceeds 90°. Side and angle labels can combine, for example “isosceles right” for a 45° 45° 90° triangle.

Common mistakes

• Pairing a side with the wrong opposite angle
• Entering three angles with no side (infinite similar triangles)
• Ignoring SSA ambiguity and accepting only one sketch
• Using degrees in a calculator set to radians
• Accepting side lengths that break the triangle inequality

Practical scenarios

Students use SSS and SAS homework sets daily. Surveyors and builders use ASA and AAS when angles are measured with a transit. Navigation problems often produce SSA. Graphic designers and game developers use side and angle constraints when laying out meshes.

Pair this tool with the Quadratic Formula Calculator when a cosine law step creates a quadratic, or the Percent Calculator when scaling similar triangle ratios.

When this calculator is useful

Use it to find missing sides and angles, check homework, compute area and perimeter, classify triangle type, and visualize a scaled diagram before drafting.

When this calculator may not be enough

It solves plane Euclidean triangles only. Spherical trigonometry, non Euclidean geometry, and 3D solid face solving need other methods. Floating point rounding can show tiny errors near 0° or 180°.

How to interpret your result

Hero text names the case and type. Metrics list all sides and angles. Heights are altitudes to each side. If SSA yields two triangles, use the solution tabs. Steps describe the main identities used.

Limitations

Results assume positive side lengths and angles between 0° and 180° that can form a triangle. Congruence cases AAA are not supported because AAA determines shape only, not size.

Sources & References

• OpenStax Algebra and Trigonometry: Non right triangles, Law of Sines: openstax.org
• OpenStax Algebra and Trigonometry: Law of Cosines: openstax.org