Triangle Calculator: Simple and Clear Math Tool

Calculate all properties of a triangle by entering known values. Select your calculation method below based on what information you have.

How to Use This Calculator

1. Select the type of information you know about your triangle from the dropdown menu
2. Enter the known values in the input fields that appear
3. Click "Calculate" to solve for all unknown values
4. View the complete results including all sides, angles, area, and perimeter

Select Calculation Method

Which rule solves which case

A triangle has six measurements (three sides, three angles), and you need exactly three of them — with at least one side — to fully determine the rest. Which formula applies depends on what you know: three sides (SSS) requires the law of cosines to find angles; two sides and the included angle (SAS) uses the law of cosines to find the third side, then the law of sines for the remaining angles; two angles and a side (ASA/AAS) uses the law of sines directly; two sides and a non-included angle (SSA) is the "ambiguous case," where the given information can sometimes match two different triangles or none at all.

Worked example: SSS with sides 7, 8, 9

Enter side a = 7, side b = 8, side c = 9 (a valid triangle, since 7+8 > 9). The law of cosines finds angle A: cos(A) = (b² + c² - a²)/(2bc) = (64 + 81 - 49)/(2×8×9) = 96/144 = 0.6667, so A = arccos(0.6667) ≈ 48.19°. The same formula rearranged for angle B: cos(B) = (a² + c² - b²)/(2ac) = (49 + 81 - 64)/(2×7×9) = 66/126 = 0.5238, so B ≈ 58.41°. Angle C is whatever's left: 180° − 48.19° − 58.41° ≈ 73.40°. Area comes from Heron's formula using the semi-perimeter s = (7+8+9)/2 = 12: Area = √(12 × (12-7) × (12-8) × (12-9)) = √(12 × 5 × 4 × 3) = √720 ≈ 26.833 square units. Perimeter is simply 7+8+9 = 24.

Why SSA can return no answer or two answers

When you know two sides and an angle that isn't between them, the given angle can sometimes be satisfied by swinging the unknown side to two different lengths — geometrically, a circle of the given side length can intersect the triangle's other ray at zero, one, or two points. This calculator returns a "no solution" alert if the sine ratio comes out greater than 1 (geometrically impossible) or if the resulting third angle is zero or negative — it reports one valid solution rather than enumerating both possibilities when the ambiguous case has two.

Reading the diagram

The canvas below the results scales the triangle so its longest side fits within 200 pixels, then plots vertices A, B, and C using the calculated angle at A to position the third point — a quick visual sanity check that the shape looks like what you expected rather than a precise-to-scale technical drawing.