Triangle Calculator

Visual triangle solver

Place three known measurements

Choose what you know, then enter each value directly beside its side or angle.

Verified geometryFormula 1.0.0 · degrees · unique triangles only

Solved triangle

Right Scalene triangle

SSS · Law of Cosines · angles in degrees

Area6

Proportional diagram using the solved side lengths.

Area: 6 square units
Perimeter: 12 units

Opposite pairsSide ↔ angle

Side a ↔ Angle A: a 3A 36.869898°
Side b ↔ Angle B: b 4B 53.130102°
Side c ↔ Angle C: c 5C 90°

Triangle-data navigator

Match the known sides and angles to a solvable triangle case

A triangle solver does not begin with a favorite formula. It begins by inventorying the measurements, pairing each side with its opposite angle, and deciding whether the data determine zero, one, or more than one triangle.

Step-by-step guide · Triangle-data navigator

A shoreline survey demonstrates an SAS triangle from field notes

Angle total: A + B + C = 180°
Existence test: each pair of sides sums above the third
Unsupported ambiguous case: SSA can yield 0, 1, or 2 triangles

Solve the opposite side first
Apply a²=b²+c²−2bc cos A. Substitution gives a²=120²+85²−2(120)(85)cos47°, so a is approximately 87.82 meters. Keeping the cosine in degree mode is essential.
Recover another angle without ambiguity
With all sides now known, use cos B=(a²+c²−b²)/(2ac). An inverse cosine gives B≈87.94°. The cosine route avoids choosing the wrong supplementary branch.
Close the triangle and calculate area
C=180°−A−B≈45.06°. The area can be found directly from the original SAS data: K=bc sin A/2≈3,730 square meters. Small differences may appear if intermediate values were rounded.

Match the known sides and angles to a solvable triangle case: Two measured shoreline segments from a marker are 120 meters and 85 meters, with an included angle of 47°. Call them b=120, c=85, and A=47° so the unknown crossing a is opposite the measured angle.

A side belongs opposite its matching capital-letter angle

In standard notation, side a lies opposite angle A, b opposite B, and c opposite C. That pairing matters more than where the triangle is drawn on screen. Two angles immediately determine the third through the 180-degree sum; three sides must pass the triangle inequality; and at least one known side is needed to fix scale. Three angles alone describe shape but allow infinitely many similar sizes.

Choose the method from the known pattern. This workbench supports SSS, SAS, ASA, and AAS. SSA is intentionally not an input mode because inverse sine returns one principal angle even when a second supplementary angle could also fit; treat it as an ambiguous-case warning for manual work. A right triangle with two known sides is a simpler special case for the Pythagorean theorem calculator.

A shoreline survey demonstrates an SAS triangle from field notes

Two measured shoreline segments from a marker are 120 meters and 85 meters, with an included angle of 47°. Call them b=120, c=85, and A=47° so the unknown crossing a is opposite the measured angle.

Solve the opposite side first
Apply a²=b²+c²−2bc cos A. Substitution gives a²=120²+85²−2(120)(85)cos47°, so a is approximately 87.82 meters. Keeping the cosine in degree mode is essential.
Recover another angle without ambiguity
With all sides now known, use cos B=(a²+c²−b²)/(2ac). An inverse cosine gives B≈87.94°. The cosine route avoids choosing the wrong supplementary branch.
Close the triangle and calculate area
C=180°−A−B≈45.06°. The area can be found directly from the original SAS data: K=bc sin A/2≈3,730 square meters. Small differences may appear if intermediate values were rounded.

Triangle validity is more than getting three positive outputs

SSS can fail before trigonometry

Lengths 2, 3, and 6 cannot close because 2+3≤6. No angle calculation can repair that geometry. If the task is only planar coverage for a valid three-side triangle, the area calculator can apply Heron’s formula without presenting every angle.

SSA can branch

Given an acute angle and its opposite side plus another side, both θ and 180°−θ may satisfy the sine ratio. Each candidate must leave a positive third angle and pass side ordering checks. This is a manual diagnostic boundary, not a supported workbench mode.

Rounded field data can nearly contradict

Measurements close to a degenerate straight line amplify numerical uncertainty. Report sensible precision and revisit the survey rather than treating a fragile computed angle as exact. The finding slope calculator answers a two-point line question, not this three-vertex closure problem.

A solved triangle deserves a geometric closing checklist

Verify that every displayed angle is positive and the three sum to 180° within the stated rounding.
Confirm that the largest side is opposite the largest angle and that no side equals or exceeds the sum of the other two.
Recalculate area by a second available route—Heron’s formula or one-half times two sides times the included sine—when the data permit.
If the desired result is solely a square-foot or square-meter surface quantity, move to the area calculator; if the geometry is a circular boundary, use the circumference calculator.

Important limitations

Triangle Calculator is limited to the finite inputs, modes, and outputs stated for the Triangle solver method. Check known measurements, validation messages, the equation, and the stated display precision before relying on area, side a, side b, and side c.

This formula passed its registered engineering tests and is published under an explicit site-owner policy authorization. No independent expert, QA, editorial, legal, or third-party review is claimed. Verify consequential results with a qualified professional.

Triangle-data navigator

Match the known sides and angles to a solvable triangle case

Step-by-step guide · Triangle-data navigator

A shoreline survey demonstrates an SAS triangle from field notes

Angle total: A + B + C = 180°
Existence test: each pair of sides sums above the third
Unsupported ambiguous case: SSA can yield 0, 1, or 2 triangles

Solve the opposite side first
Apply a²=b²+c²−2bc cos A. Substitution gives a²=120²+85²−2(120)(85)cos47°, so a is approximately 87.82 meters. Keeping the cosine in degree mode is essential.
Recover another angle without ambiguity
With all sides now known, use cos B=(a²+c²−b²)/(2ac). An inverse cosine gives B≈87.94°. The cosine route avoids choosing the wrong supplementary branch.
Close the triangle and calculate area
C=180°−A−B≈45.06°. The area can be found directly from the original SAS data: K=bc sin A/2≈3,730 square meters. Small differences may appear if intermediate values were rounded.

A side belongs opposite its matching capital-letter angle

A shoreline survey demonstrates an SAS triangle from field notes

Two measured shoreline segments from a marker are 120 meters and 85 meters, with an included angle of 47°. Call them b=120, c=85, and A=47° so the unknown crossing a is opposite the measured angle.

Solve the opposite side first
Apply a²=b²+c²−2bc cos A. Substitution gives a²=120²+85²−2(120)(85)cos47°, so a is approximately 87.82 meters. Keeping the cosine in degree mode is essential.
Recover another angle without ambiguity
With all sides now known, use cos B=(a²+c²−b²)/(2ac). An inverse cosine gives B≈87.94°. The cosine route avoids choosing the wrong supplementary branch.
Close the triangle and calculate area
C=180°−A−B≈45.06°. The area can be found directly from the original SAS data: K=bc sin A/2≈3,730 square meters. Small differences may appear if intermediate values were rounded.

Triangle validity is more than getting three positive outputs

SSS can fail before trigonometry

SSA can branch

Rounded field data can nearly contradict

A solved triangle deserves a geometric closing checklist

Verify that every displayed angle is positive and the three sum to 180° within the stated rounding.
Confirm that the largest side is opposite the largest angle and that no side equals or exceeds the sum of the other two.
Recalculate area by a second available route—Heron’s formula or one-half times two sides times the included sine—when the data permit.
If the desired result is solely a square-foot or square-meter surface quantity, move to the area calculator; if the geometry is a circular boundary, use the circumference calculator.

Triangle Calculator

Place three known measurements

Enter values on the diagram

Right Scalene triangle

Match the known sides and angles to a solvable triangle case

A shoreline survey demonstrates an SAS triangle from field notes

Solve the opposite side first

Recover another angle without ambiguity

Close the triangle and calculate area

A side belongs opposite its matching capital-letter angle

A shoreline survey demonstrates an SAS triangle from field notes

Triangle validity is more than getting three positive outputs

SSS can fail before trigonometry

SSA can branch

Rounded field data can nearly contradict

A solved triangle deserves a geometric closing checklist

Important limitations

Place three known measurements

Enter values on the diagram

Right Scalene triangle

Match the known sides and angles to a solvable triangle case

A shoreline survey demonstrates an SAS triangle from field notes

Solve the opposite side first

Recover another angle without ambiguity

Close the triangle and calculate area

A side belongs opposite its matching capital-letter angle

A shoreline survey demonstrates an SAS triangle from field notes

Triangle validity is more than getting three positive outputs

SSS can fail before trigonometry

SSA can branch

Rounded field data can nearly contradict

A solved triangle deserves a geometric closing checklist