SSS Triangle Calculator
Enter the lengths of all three sides to find angles, area, and perimeter.
Invalid triangle: The sum of any two sides must be greater than the third side.
Calculated Results
Angle A:
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Angle B:
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Angle C:
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Area:
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Perimeter:
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Semi-perimeter (s):
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Understanding the SSS (Side-Side-Side) Triangle Theorem
The SSS (Side-Side-Side) theorem states that if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent. In trigonometry, having three sides allows us to solve for all internal angles using the Law of Cosines.
How to Calculate Angles from Three Sides
To find the angles of a triangle when you only know the side lengths (a, b, and c), we use the Law of Cosines formula rearranged to solve for the angle:
- Angle A: cos(A) = (b² + c² – a²) / 2bc
- Angle B: cos(B) = (a² + c² – b²) / 2ac
- Angle C: cos(C) = (a² + b² – c²) / 2ab
The Triangle Inequality Theorem
Before calculating, it is vital to ensure that the three side lengths can actually form a triangle. The Triangle Inequality Theorem states that for any triangle, the sum of the lengths of any two sides must be strictly greater than the length of the remaining side:
| Condition | Must be True |
|---|---|
| Side A + Side B | > Side C |
| Side A + Side C | > Side B |
| Side B + Side C | > Side A |
Calculating Area with Heron's Formula
Once you have the three sides, you can find the area without knowing the height using Heron's Formula:
1. Calculate the semi-perimeter (s): s = (a + b + c) / 2
2. Calculate Area: Area = √[s(s – a)(s – b)(s – c)]
Practical Example
Suppose you have a triangle with sides a = 5, b = 6, and c = 7.
- Perimeter: 5 + 6 + 7 = 18
- Semi-perimeter (s): 18 / 2 = 9
- Angle A: arccos((6² + 7² – 5²) / (2 * 6 * 7)) ≈ 44.4°
- Area: √[9(9-5)(9-6)(9-7)] = √[9 * 4 * 3 * 2] = √216 ≈ 14.7 units²