Calculate all three sides of a triangle given three angles and one known side using the law of sines. Supports AAS, ASA, and area-based modes. Shows triangle type, perimeter, area, and altitude.
The triangle sides calculator uses the law of sines to find all three side lengths when you know the three angles and at least one side. The law of sines states that in any triangle, the ratio of a side to the sine of its opposite angle is constant: a/sin A = b/sin B = c/sin C. This fundamental relationship allows you to solve for unknown sides whenever you have sufficient angle-side information.
The page supports two input modes. In the standard AAS/ASA mode, you enter all three angles (which must sum to 180°) and one known side, and the calculator finds the remaining two sides from that setup. In the area-based mode, you enter the area and two angles, and the page back-calculates the sides that produce that area.
Beyond the raw side lengths, the calculator reports the full suite of triangle properties: perimeter, semi-perimeter, area (via the formula ½ab sin C), all three altitudes, the circumradius R = a/(2 sin A), and the inradius r = Area/s. It classifies the triangle by its sides (equilateral, isosceles, or scalene) and by its angles (acute, right, or obtuse).
Presets let you quickly load common triangles — equilateral, 30-60-90, 45-45-90, and more — and a reference table compares classic triangle types side by side. Whether you are studying trigonometry, surveying land, or designing structures, the page gives you the full triangle property set from the same angle-side input.
When you know the triangle's angle pattern and one side length, the real challenge is translating that information into a consistent scale for all three sides. The law of sines does that elegantly, but it is still easy to mix up which side is opposite which angle or to lose track of the common ratio. This calculator makes that conversion immediate and keeps the opposite-side relationships visible.
The area-plus-angles mode is also useful because it reverses the usual problem structure. Instead of starting from one side, you can start from the triangle's size and shape together, then recover the side lengths that match both conditions. That gives the tool value beyond standard AAS and ASA exercises.
Law of sines: a/sin A = b/sin B = c/sin C
Area: ½ × a × b × sin C
Perimeter: P = a + b + c
Semi-perimeter: s = P / 2
Altitude to side a: hₐ = 2 × Area / a
Circumradius: R = a / (2 × sin A)
Inradius: r = Area / sResult: Side b ≈ 8.66, Side c = 10, Perimeter ≈ 23.66, Area ≈ 21.65
In a 30-60-90 triangle with side a = 5 (opposite 30°): By the law of sines, a/sin 30° = b/sin 60° = c/sin 90°. So 5/0.5 = 10, b = 10 × sin 60° ≈ 8.66, c = 10 × sin 90° = 10. Area = ½ × 5 × 8.66 × sin 90° ≈ 21.65.
If all three angles of a triangle are known, the shape is fixed but the actual side lengths are not. You still need one piece of scale information, such as a known side or an area value, to determine the triangle completely. That is why this calculator separates shape data from size data and combines them through the law of sines.
The law of sines works only when each side is matched with its opposite angle correctly. In a 30-60-90 triangle, the shortest side must sit opposite 30° and the longest side opposite 90°. Those relative positions are more important than the side labels themselves, and checking them early prevents many setup errors.
Area plus angles is a more advanced input mode because it asks you to reason backward from both shape and total size. Once one side is recovered, the remaining sides follow from the same sine ratio. This is a good way to practice seeing the triangle as a connected system rather than as three independent side calculations.
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The law of sines states that a/sin A = b/sin B = c/sin C. It relates each side of a triangle to the sine of its opposite angle, and all three ratios are equal.
If you know two angles, you can compute the third (since they sum to 180°), but you still need at least one side length to determine the actual sizes. Without a side, you can only find the shape, not the scale.
Use the law of sines when you know an angle-side pair (AAS, ASA). Use the law of cosines when you know three sides (SSS) or two sides + included angle (SAS).
The circumradius R is the radius of the circle that passes through all three vertices. It equals a / (2 sin A) for any side a and its opposite angle A.
Any triangle where all angles are positive and sum to 180°. It handles acute, right, and obtuse triangles, as well as equilateral, isosceles, and scalene ones.
Using the formula Area = ½ × a × b × sin C, where C is the angle between sides a and b. Alternatively, once all sides are known, Heron's formula can also be used.
Calculate all three angles of a triangle from three side lengths using the law of cosines. Shows angle classification, radian conversion, sum verification, area, and full triangle properties.
Calculate triangle area, perimeter, angles, altitudes, circumradius, inradius, and classification. Supports base-height, Heron's formula, and SAS methods with unit selection and a complete 15-prop..
Solve any right triangle from two known values — legs, hypotenuse, or angles. Computes all sides, angles, area, perimeter, inradius, circumradius, and altitude. Includes Pythagorean triple presets.