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Triangle Calculator | Solve Sides, Angles, Area & Perimeter

Q: How do surveyors measure an unreachable distance?

They measure a baseline and two angles, then let the law of sines do the rest. From a 100 m baseline with adjacent angles 50° and 60°, the third angle is 70° and the far side is 100·sin(60°)/sin(70°) ≈ 92.2 m. One measured side plus two angles fully determines the triangle.

Q: What is Heron's formula and when is it used?

Heron's formula finds area from three sides alone: area = √(s(s−a)(s−b)(s−c)) where s is the semiperimeter. For sides 5, 12, 13: s = 15, area = √(15×10×3×2) = √900 = 30. It is the SSS-case workhorse when no height or angle is known.

Solve any triangle from three sides, two sides and an angle, or two angles and a side — the tool returns all missing sides, angles, area and perimeter. Right triangles use the Pythagorean theorem and oblique triangles use the sine and cosine laws, with full working shown.

Last updated: June 2026 · Free · No sign-up required

Results

Enter values above and click Calculate to see your result instantly.

Quick reference

Need	Formula
Area from base/height	A = 1/2bh
Right triangle hypotenuse	c = √(a²+b²)
Perimeter	a+b+c
Angle sum	A+B+C = 180°

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For students and teachers, the final number is only half the value. Every math tool here exposes the formula it applied, the intermediate steps, and the rounding rule, so you can follow along, check your homework, or use the answer in a proof or report with confidence.

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Below the calculator we link a small set of plain-English explainer pages — "What is a percentage?", "Why does PEMDAS matter?", and so on. They cover the underlying concept in 4–6 short paragraphs. Read those before the calculator if the topic is new, or after if you want the extra context.

Interpretation guide

Triangle type	Clue
Right	One 90° angle
Isosceles	Two equal sides
Equilateral	Three equal sides and 60° angles

lightbulb Worked example: a 5-12-13 garden plot

A triangular garden has sides 5 m, 12 m, and 13 m. Find the angles, area, and perimeter. Check for a right angle: 5² + 12² = 25 + 144 = 169 = 13² — a Pythagorean triple, so the angle opposite the 13 m side is 90°. The acute angles: arcsin(5/13) ≈ 22.62° and arcsin(12/13) ≈ 67.38°. Area = ½ × 5 × 12 = 30 m² (Heron check: s = 15, √(15×10×3×2) = √900 = 30). Perimeter = 5 + 12 + 13 = 30 m.

Result: Angles ≈ 22.62°, 67.38°, 90°; area 30 m²; perimeter 30 m.

What this means: Recognizing the Pythagorean triple confirms the right angle without any trigonometry. Thirty square meters of sod and 30 m of edging follow directly — a tidy case where the area formula and Heron's formula check each other.

Formula & methodology

Formula: Area = 1/2bh; a² + b² = c² for right triangles

How triangles are solved

Any valid combination of sides and angles determines the rest. The solver picks its method from your inputs:

Right triangles: Pythagorean theorem a² + b² = c², plus inverse trig for the acute angles.
Two angles + a side (AAS/ASA): third angle = 180° − A − B, then the law of sines a/sin A = b/sin B = c/sin C.
Two sides + included angle (SAS), or three sides (SSS): law of cosines c² = a² + b² − 2ab·cos C; area via Heron's formula √(s(s−a)(s−b)(s−c)).
Inputs are checked against the triangle inequality first — impossible side sets are rejected.

Reading the result

You get all three sides, all three angles, area, and perimeter. A 5-12-13 input resolves to angles ≈ 22.62°, 67.38°, 90°, area 30, perimeter 30 — the solver recognizes the Pythagorean triple.

Authoritative source: Wolfram MathWorld

Frequently asked questions

How do I use the triangle calculator?

Enter any combination that defines the triangle — three sides, two sides and the included angle, or two angles and one side — and the tool solves for the rest. It uses the Pythagorean theorem a² + b² = c² for right triangles and the standard area formula ½ × base × height. For a 3-4-5 right triangle the hypotenuse is √(9 + 16) = 5 and the area is 6.

How do you solve a 5-12-13 right triangle?

The angles come from inverse trig: arcsin(5/13) ≈ 22.62° and arcsin(12/13) ≈ 67.38°, with the right angle 90°. Area = ½ × 5 × 12 = 30, perimeter = 5 + 12 + 13 = 30. The solver detects the Pythagorean triple (5² + 12² = 13²) and confirms the right angle.

When do you need the law of sines versus the law of cosines?

Law of sines (a/sin A = b/sin B) fits AAS and ASA cases — two angles plus a side. Law of cosines (c² = a² + b² − 2ab·cos C) fits SAS and SSS — two sides with the included angle, or three sides. The solver picks the right law from whichever combination you enter.

Why do my results differ from another calculator?

The Triangle Calculator is free, private, and accurate: it runs entirely in your browser (no uploads, no accounts), applies the standard calculation, and explains each step so you can verify the result. Enter any combination of sides and angles — three sides, two sides and an included angle, or two angles and a side — and this triangle solver returns the remaining sides, all three angles, the area, and the perimeter. It applies the law of sines and law of There is no limit on usage, and it works on any device.

Can three sides fail to make a triangle?

Yes, when they violate the triangle inequality: each side must be shorter than the sum of the other two. Sides 2, 3, and 10 cannot form a triangle because 2 + 3 < 10. The solver checks this first and reports impossible inputs instead of computing nonsense angles.

How do surveyors measure an unreachable distance?

They measure a baseline and two angles, then let the law of sines do the rest. From a 100 m baseline with adjacent angles 50° and 60°, the third angle is 70° and the far side is 100·sin(60°)/sin(70°) ≈ 92.2 m. One measured side plus two angles fully determines the triangle.

What is Heron's formula and when is it used?

Heron's formula finds area from three sides alone: area = √(s(s−a)(s−b)(s−c)) where s is the semiperimeter. For sides 5, 12, 13: s = 15, area = √(15×10×3×2) = √900 = 30. It is the SSS-case workhorse when no height or angle is known.

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