All Three Angles from Three Sides

Solving the cosine rule for the angle gives all three from the sides alone. It is how you find angles with a tape measure and no protractor, which is why carpenters and surveyors use it. Check that the three add to 180°.

Your numbers

Side a(cm)Side b(cm)Side c(cm)

Angle A

35.43°

Angle B

48.19°

Angle C

96.38°

What it means

The largest angle is 96.4° — an obtuse triangle.

Formula

Angle A = acos((Side b ² + Side c ² − Side a ²) ÷ (2 × Side b × Side c))

If one side exceeds the other two combined, no triangle exists and the angles collapse to 0 or 180°.

What to enter

Input	Default	Accepted range
Side a (cm)The length of the first side.	7	0.01 and up
Side b (cm)The length of the second side.	9	0.01 and up
Side c (cm)The third side; it must be shorter than the other two combined.	12	0.01 and up

Step by step

FormulaAngle A = acos((Side b ² + Side c ² − Side a ²) ÷ (2 × Side b × Side c))

With the default numbersAngle A = acos((9 ² + 12 ² − 7 ²) ÷ (2 × 9 × 12))

AnswerAngle A = 35.43 °

Quick reference table

Results when only Side a (cm) changes and everything else stays put.

Side a (cm)	Angle A (°)	Angle B (°)	Angle C (°)
3.5	9.95	26.38	143.66
5.25	23.93	44.05	112.02
7	35.43	48.19	96.38
10.5	57.91	46.57	75.52
14	82.28	39.57	58.14

What each result means

Result	At default values
Angle A (°)The angle opposite side a.	35.43
Angle B (°)The angle opposite side b.	48.19
Angle C (°)The angle opposite side c; the three add to 180°.	96.38

Common mistakes

If one side exceeds the other two combined, no triangle exists and the angles collapse to 0 or 180°.

Glossary

Side a: The length of the first side.
Side b: The length of the second side.
Side c: The third side; it must be shorter than the other two combined.
Angle A: The angle opposite side a.
Angle B: The angle opposite side b.
Angle C: The angle opposite side c; the three add to 180°.

Frequently asked questions

QHow is All Three Angles from Three Sides calculated?

Angle A = acos((Side b ² + Side c ² − Side a ²) ÷ (2 × Side b × Side c)) — Solving the cosine rule for the angle gives all three from the sides alone. It is how you find angles with a tape measure and no protractor, which is why carpenters and surveyors use it. Check that the three add to 180°.

QCan you walk through an example?

With Side a 7cm, Side b 9cm, Side c 12cm, the answer is Angle A 35.43°.

QWhat do I need to enter?

Enter Side a, Side b, Side c. The result recalculates as you type, and an empty box counts as zero.

QHow much does the answer move if I change a number?

Changing only Side a (cm) moves the answer to Side a (cm) 3.5 → Angle A (°) 9.95 and Side a (cm) 14 → Angle A (°) 82.28. The table below lays out five steps.

QHow are the numbers rounded?

Money is shown to the nearest whole unit, percentages to one decimal place and everything else to two. What you see is rounded; the calculation itself carries the unrounded value forward.

QAnything to watch out for?

If one side exceeds the other two combined, no triangle exists and the angles collapse to 0 or 180°.

Related calculators

Pythagorean TheoremFind the hypotenuse from the two legs of a right triangle.Missing Leg of a Right TriangleFind the remaining side from the hypotenuse and one leg.Angle from Two SidesGet the angle of a slope from its rise and run.Trig Function ValuesEnter an angle and see sine, cosine and tangent at once.

Keep your length units consistent across inputs — mixing centimetres and metres throws results far off.