How do you find the angles of a triangle when you know all three sides?

Use the law of cosines. For a triangle with sides a, b, and c, angle A (opposite side a) is found with: A = arccos((b² + c² - a²) / (2bc)). Angles B and C are found similarly by cycling the letters. Once you have two angles, the third is simply 180° minus the sum of the other two.

What is Heron's formula for triangle area?

Heron's formula calculates the area of a triangle from its three side lengths without needing the height. First compute the semi-perimeter: s = (a + b + c) / 2. Then the area is: Area = sqrt(s × (s - a) × (s - b) × (s - c)). For example, a triangle with sides 5, 6, and 7 has s = 9 and area = sqrt(9 × 4 × 3 × 2) = sqrt(216) ≈ 14.70 square units.

When do three side lengths NOT form a valid triangle?

Three side lengths form a valid triangle only if each side is strictly less than the sum of the other two. This is called the triangle inequality. For example, sides 1, 2, and 5 do not form a triangle because 1 + 2 = 3, which is less than 5. The calculator will warn you if the entered sides cannot form a real triangle.

ABC Triangle Calculator (SSS)

Enter all three side lengths of a triangle (a, b, and c) to instantly find every angle, the area, the perimeter, and the triangle type. This is the SSS (Side-Side-Side) case, solved using the law of cosines and Heron's formula.

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Standard geometry Law of cosines and Heron's formula. Exact results to 3 decimal places.

1. Enter the Three Sides

Side a (opposite angle A)

units

Side b (opposite angle B)

units

Side c (opposite angle C)

units

2. Results

These three side lengths cannot form a valid triangle. Each side must be less than the sum of the other two (triangle inequality).

Angle A (opposite side a)-

Angle B (opposite side b)-

Angle C (opposite side c)-

Perimeter-

Semi-perimeter (s)-

Area-

Triangle type (angles)-

Triangle type (sides)-

Summary

Angle A

degrees

Angle B

degrees

Angle C

degrees

Area
-
square units

Worked Example

Inputs: a = 5, b = 6, c = 7
Perimeter: 5 + 6 + 7 = 18 units
Semi-perimeter (s): 18 / 2 = 9
Area (Heron's formula): sqrt(9 x (9 - 5) x (9 - 6) x (9 - 7)) = sqrt(9 x 4 x 3 x 2) = sqrt(216) = 14.697 sq units
Angle A (law of cosines): arccos((6² + 7² - 5²) / (2 x 6 x 7)) = arccos(60 / 84) = 44.415°
Angle B: arccos((5² + 7² - 6²) / (2 x 5 x 7)) = arccos(38 / 70) = 57.122°
Angle C: arccos((5² + 6² - 7²) / (2 x 5 x 6)) = arccos(12 / 60) = 78.463°
Check: 44.415 + 57.122 + 78.463 = 180.000°
Type: Scalene (all sides different), Acute (all angles less than 90°)

How to Solve an ABC Triangle (SSS)

When you know all three side lengths of a triangle, you can find everything else about it using two key formulas.

The Law of Cosines

The law of cosines extends Pythagoras' theorem to any triangle. To find angle A (the angle opposite side a):

A = arccos((b² + c² - a²) / (2bc))

Repeat the formula, cycling through the sides, to find angles B and C. This always works as long as the three sides satisfy the triangle inequality (each side must be less than the sum of the other two).

Heron's Formula for Area

Heron's formula gives the area of a triangle from its three sides alone, without needing the height:

Calculate the semi-perimeter: s = (a + b + c) / 2
Calculate the area: Area = sqrt(s x (s - a) x (s - b) x (s - c))

Triangle Types

Classification	By sides	By angles
Equilateral	All three sides equal	All angles 60°
Isosceles	Two sides equal	Two angles equal
Scalene	All sides different	All angles different
Acute	-	All angles less than 90°
Right	-	One angle exactly 90°
Obtuse	-	One angle greater than 90°

Triangle Inequality

Three lengths can only form a valid triangle if each length is strictly less than the sum of the other two. For example, sides of 1, 2, and 10 cannot form a triangle because 1 + 2 = 3, which is less than 10. The calculator will alert you if the sides you enter fail this test.

Related Calculators

Method: Angles calculated using the law of cosines: A = arccos((b² + c² - a²) / (2bc)). Area calculated using Heron's formula: Area = sqrt(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2. Triangle inequality check: each side must be less than the sum of the other two.

Results are rounded to three decimal places for display. The calculator works with any consistent unit of length (cm, m, inches, etc.).

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