What does SSS, SAS, ASA and AAS mean for triangles?

These abbreviations describe which three values you know. SSS means all three sides. SAS means two sides and the angle between them. ASA means two angles and the included side. AAS means two angles and a non-included side. Any combination fully determines the triangle.

How do you find the area of a triangle from its sides?

Use Heron's formula: calculate the semi-perimeter s = (a + b + c) / 2, then area = square root of s(s-a)(s-b)(s-c). If you know two sides and the included angle, use area = 0.5 times a times b times sin(C).

What is the law of cosines?

The law of cosines states c squared = a squared + b squared minus 2ab times cos(C). It generalises Pythagoras for any triangle and lets you find a side or angle when you know the other three values.

Triangle Calculator

This triangle calculator solves any triangle when you supply three known values, whether those are sides, angles, or a mix of both. A triangle is fully determined by any of four classic combinations: all three sides (SSS), two sides and the included angle (SAS), two angles and the included side (ASA), or two angles and a non-included side (AAS). From those three inputs the calculator works out every remaining measurement using the law of cosines and the law of sines, the two fundamental theorems of triangle geometry. The law of cosines states that c squared equals a squared plus b squared minus two times a times b times the cosine of angle C, which lets you recover a side or angle in any triangle regardless of whether it contains a right angle. The law of sines states that the ratio of each side to the sine of its opposite angle is constant across all three pairs, which resolves the remaining unknowns. Once all three sides and angles are known, the area follows from Heron's formula using the semi-perimeter. Enter the three values you know across the six input fields, leaving the others blank, and the calculator returns all six measurements, the area, the perimeter, and a classification of the triangle as acute, right or obtuse, and as equilateral, isosceles or scalene. The default values are a=3, b=4, c=5, the classic 3-4-5 right triangle used in countless geometry problems. Results are for reference only.

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Side a

Side b

Side c

Angle A (degrees) opposite side a

Angle B (degrees) opposite side b

Angle C (degrees) opposite side c

6.00

area (square units)

Perimeter12.00

Angle A36.87°

Angle B53.13°

Angle C90.00°

TypeRight / Scalene

How it works

The calculator counts which fields you have filled in to determine the case. For SSS it uses the law of cosines to derive each angle: cos(A) = (b² + c² − a²) / (2bc). For SAS it finds the missing side via the law of cosines then the remaining angles via the law of sines. For ASA and AAS it first calculates the third angle (the three must sum to 180°), then applies the law of sines to recover the two unknown sides. Area is computed with Heron's formula: s = (a+b+c)/2, area = √(s(s−a)(s−b)(s−c)). The type checks whether any angle equals 90° (right), exceeds 90° (obtuse), or all are below 90° (acute), and whether all sides match (equilateral), two match (isosceles), or all differ (scalene).

Worked example

With the default values a=3, b=4, c=5, the law of cosines gives angle A = arccos((16+25−9)/(40)) = 36.87°. Angle B = arccos((9+25−16)/(30)) = 53.13°. Angle C = 180 − 36.87 − 53.13 = 90.00°. Perimeter = 12.00. Heron's formula gives s=6, area = √(6×3×2×1) = 6.00 square units. The triangle is right and scalene, confirming the classic 3-4-5 result.

Related calculators

Heron's Formula Calculator: find triangle area from three sides using the semi-perimeter method.
Hypotenuse Calculator: apply Pythagoras's theorem to right triangles specifically.
Law of Sines and Cosines Calculator: explore the underlying rules with deeper worked examples.
Area Calculator: compute the area of circles, rectangles, and other common 2D shapes.

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