Triangle Calculator

Tool Overview

This tool is a triangle calculator. You enter exactly three values: sides (a, b, c) or angles (A, B, C) or a mix. The tool figures out which case you have and computes the missing sides, angles, and area. It draws the triangle and shows the steps it used.

Many people need to find missing sides or angles of a triangle for school, work, or projects. Doing this by hand means choosing the right rule (Law of Sines, Law of Cosines, or Heron) and doing several steps. One mistake can give the wrong answer. This tool does the math for you. It supports three sides (SSS), two sides and the included angle (SAS), two angles and a side (ASA or AAS), and two sides and a non-included angle (SSA). For SSA it can show that no triangle exists, one triangle exists, or two triangles exist. You get the sides, angles, area, a drawing, and a step-by-step explanation. You can copy the results or try example inputs.

The tool is for students learning geometry, teachers preparing examples, and anyone who needs quick triangle solutions. You do not need to memorize formulas. If you can enter three known values in the right boxes, you can use it.

Background and Concept Explanation

A triangle has three sides and three angles. The angles add up to 180 degrees. If you know three pieces of information (sides or angles), you can usually find the rest. The way you find them depends on what you know. Three sides (SSS) use the Law of Cosines to get angles, then area from Heron. Two sides and the angle between them (SAS) use the Law of Cosines to get the third side, then the other angles. Two angles and a side (ASA or AAS) use the fact that angles sum to 180 to get the third angle, then the Law of Sines to get the other sides. Two sides and an angle not between them (SSA) are trickier: sometimes no triangle fits, sometimes one, sometimes two. People often get SSA wrong by hand because they forget to check the height and the ambiguous case. This tool handles all these cases and shows you the steps so you can follow the logic.

Key Features

• Six input fields. You have three side boxes (a, b, c) and three angle boxes (A, B, C in degrees). You must fill exactly three values. You can use three sides, two sides and one angle, or two angles and one side. The tool detects which case you have. This keeps one simple form for all cases.
• Four solution types. The tool supports SSS (three sides), SAS (two sides and included angle), ASA or AAS (two angles and a side), and SSA (two sides and non-included angle). Each type uses the right formulas. For SSA the tool may return 0, 1, or 2 solutions and explains why.
• Validation. Side values must be greater than 0. Angle values must be greater than 0 and less than 180. The tool checks that the three sides satisfy the triangle inequality (sum of any two sides greater than the third) for SSS. If you enter more than three values, it asks you to use exactly three. So invalid inputs are caught before calculation.
• Result: type and solutions. The result shows the case name (SSS, SAS, ASA, AAS, or SSA). It lists one or two solutions. Each solution has sides a, b, c; angles A, B, C in degrees; and area. Sides and angles are shown to four decimal places; area in sq units. So you see all missing values in one place.
• Triangle drawing. The tool draws the triangle to scale. For SSA with two solutions it can show the height. You can switch between Triangle 1 and Triangle 2 when two solutions exist. So you see the shape that matches the numbers.
• Logic and steps. A section lists the steps the tool used (e.g. Law of Cosines to find angle A, then angle C = 180 - A - B). For the SSA ambiguous case it explains why two triangles exist. So you can learn or check the method.
• Copy results. A button copies the result text (case, sides, angles, area, and steps) to your clipboard. You can paste it into a document or another app.
• Try examples. Four buttons load sample inputs: SSS (three sides), SAS (two sides and angle), ASA (two angles and side), SSA (ambiguous case). Click one to fill the form and run the calculation. Helpful when you are learning the cases.
• Reset. A button clears all inputs and the result. You can then enter new values.

Common Use Cases

• Three sides known (SSS). Enter the lengths of a, b, and c. Use this when you measured all three sides. The tool finds the three angles and the area. Good for land or construction when you have side lengths.
• Two sides and the angle between them (SAS). Enter two sides and the angle that is between those two sides. The tool finds the third side and the other two angles and the area. Common in surveying and design.
• Two angles and a side (ASA or AAS). Enter two angles and any one side. The tool finds the third angle (angles sum to 180), then the other two sides and the area. Useful when you know angles from a diagram and one side length.
• Two sides and an angle not between them (SSA). Enter two sides and the angle opposite one of them. The tool checks if no triangle, one triangle, or two triangles fit and shows the result(s). Use this when you have a side-side-angle layout and need to see all possibilities.
• Check homework. Enter the same three values as in your problem. Compare the tool result and steps with your answer. The step list helps you see if you used the right rule.
• Get area from sides. For SSS the tool uses Heron formula to compute area. You get area without entering angles. For other cases area is computed from the solved sides.

How to Use This Tool (Step-by-Step)

1. Open the tool. You see six input fields: sides a, b, c and angles A, B, C. Below them are Try Examples (SSS, SAS, ASA, SSA), Calculate Solution, and Reset.
2. Enter exactly three values. Type numbers in the boxes for the sides or angles you know. Leave the rest empty. Use decimals if needed. Sides must be greater than 0; angles must be between 0 and 180 (not 180). For SSS the three sides must satisfy the triangle inequality.
3. Click Calculate Solution. The tool detects the case (SSS, SAS, ASA/AAS, or SSA) and computes the missing sides, angles, and area. If something is invalid (e.g. sides that cannot form a triangle), you see an error message instead of a result.
4. Read the result. The header shows the case name. Below you see the triangle drawing and the Computed Properties (sides, angles, area). For SSA with two solutions you see Triangle 1 and Triangle 2 buttons; click one to switch the drawing and values.
5. Read the Logic and Steps section to see how the tool got the answer. For SSA ambiguous case it explains why two triangles exist.
6. Optional: click Copy to copy the full result text to your clipboard.
7. Optional: click one of the example buttons (SSS, SAS, ASA, SSA) to load sample inputs and run that case.
8. To start over, click Reset. All fields clear and you can enter new values.

Calculations and Logic

The tool needs exactly three known values (sides or angles). It counts how many sides and how many angles you gave and picks the right method.

SSS (three sides). It checks the triangle inequality: a+b>c, a+c>b, b+c>a. If that fails, no triangle exists. Otherwise it uses the Law of Cosines to find two angles: A = arccos((b² + c² - a²) / (2bc)), B = arccos((a² + c² - b²) / (2ac)), then C = 180 - A - B. Area is from Heron: s = (a+b+c)/2, area = √(s(s-a)(s-b)(s-c)).

SAS (two sides and included angle). It uses the Law of Cosines to find the third side: side³ = side1² + side2² - 2·side1·side2·cos(angle). Then it finds the other two angles with the Law of Cosines and the fact that angles sum to 180. Area is from Heron using the three sides.

ASA or AAS (two angles and a side). It finds the third angle (180 minus the sum of the two given angles). Then it uses the Law of Sines: the ratio side/sin(opposite angle) is the same for all three. So it computes that ratio from the known side and angle, then gets the other two sides. Area is from Heron.

SSA (two sides and non-included angle). It computes the height h = adjacent_side × sin(given_angle). If the angle is 90 or more: no solution if opposite_side ≤ adjacent_side; one solution otherwise. If the angle is less than 90: no solution if opposite_side < h; one solution (right triangle) if opposite_side ≈ h; one solution if opposite_side ≥ adjacent_side; two solutions (ambiguous case) if h < opposite_side < adjacent_side. For one or two solutions it uses the Law of Sines to find the missing angle(s) and side(s). Area is from Heron. All angles are in degrees; the tool uses radians internally for sin and cos.

Reference: Cases and What You Enter

Rules: Exactly three values. Sides > 0. Angles > 0 and < 180°. For SSS, sum of any two sides must be greater than the third.

Tips, Limitations and Best Practices

Enter exactly three values. If you enter more than three the tool asks you to use exactly three. Leave the unknown fields empty. Use the same units for all sides (e.g. all in meters); the tool does not convert units. Angles are in degrees only.

For SSS, check that your three sides can form a triangle: each side must be less than the sum of the other two. For SSA, if you get no solution it may be because the given side opposite the angle is too short (shorter than the height). Try the example buttons to see valid SSA inputs that give one or two solutions.

The tool does not compute perimeter explicitly; you can add the three side lengths yourself from the result. It does not support entering area as one of the three inputs. All angles are in degrees; there is no radian mode. The drawing is to scale but may be rotated; side labels a, b, c correspond to the opposite angles A, B, C. Use the Logic and Steps section to verify the method and to learn how each case is solved.