Triangle Side Calculator: Find Sides Using Angles

Your expert tool for calculating the sides of a triangle when you know two angles and one side.

Triangle Solver

Triangle Properties Summary

All calculated properties of the triangle.

Property	Value
Angle A	—
Angle B	—
Angle C	—
Side a	—
Side b	—
Side c	—
Area	—
Perimeter	—

Deep Dive into Calculating Sides of a Triangle Using Angles

What is Calculating Sides of a Triangle Using Angles?

Calculating the sides of a triangle using angles is a fundamental concept in trigonometry that allows you to determine unknown side lengths when you have a combination of known angles and at least one known side length. This process, often called “solving a triangle,” is crucial in fields like engineering, physics, surveying, and navigation. The two primary scenarios covered by this calculator are Angle-Side-Angle (ASA) and Angle-Angle-Side (AAS), both of which rely on the Law of Sines.

A common misunderstanding is that knowing only the three angles of a triangle is enough to find its side lengths. However, knowing only angles determines the triangle’s shape, but not its size. You can have an infinite number of triangles with the same angles but different side lengths (similar triangles). That’s why at least one side length is required for a unique solution.

The Formula for Calculating Sides with Angles: The Law of Sines

The primary tool for solving triangles when you know two angles and a side is the Law of Sines. It establishes a relationship between the sides of a triangle and the sines of their opposite angles. The formula is elegant and powerful.

For a triangle with angles A, B, and C, and sides a, b, and c opposite those angles respectively, the Law of Sines states:

a / sin(A) = b / sin(B) = c / sin(C)

This ratio is constant for any given triangle. To use it, you first find the third angle (since all three angles must sum to 180 degrees), and then you can set up proportions to solve for the unknown sides.

Variables Table

Variable	Meaning	Unit (Auto-inferred)	Typical Range
a, b, c	The lengths of the sides of the triangle.	Length (cm, m, inches, etc.)	> 0
A, B, C	The angles opposite sides a, b, and c.	Degrees or Radians	> 0 and < 180 degrees
sin(A), sin(B), sin(C)	The sine of each respective angle.	Unitless ratio	-1 to 1 (but 0 to 1 for triangle angles)

Practical Examples

Example 1: Surveying a River (AAS Case)

A surveyor wants to measure the width of a river. She stands at point C, sights a tree at point B directly across the river, and then walks 100 meters to point A. She measures the angle CAB to be 70 degrees. The angle at point C (ACB) is a right angle (90 degrees). How long is side ‘a’ (the width of the river)?

• Inputs: Angle A = 70°, Angle C = 90°, Side c = 100 m.
• Find Angle B: B = 180° – 90° – 70° = 20°.
• Apply Law of Sines: a / sin(70°) = 100 / sin(90°).
• Result: a = 100 * sin(70°) / sin(90°) ≈ 93.97 meters. The river is about 94 meters wide. Check your work with a geometry calculator.

Example 2: Designing a Truss (ASA Case)

An engineer is designing a triangular truss for a roof. The base of the truss (side c) is 15 feet long. The base angles are specified as Angle A = 35° and Angle B = 35° (an isosceles triangle).

• Inputs: Angle A = 35°, Angle B = 35°, Side c = 15 ft.
• Find Angle C: C = 180° – 35° – 35° = 110°.
• Apply Law of Sines: a / sin(35°) = 15 / sin(110°).
• Result: a = 15 * sin(35°) / sin(110°) ≈ 9.16 feet. Since it’s isosceles, side b will also be 9.16 feet. You can explore more about this with a AAS triangle calculator.

How to Use This Triangle Side Calculator

1. Enter Known Angles: Input the two known angles of your triangle into the ‘Angle A’ and ‘Angle B’ fields. Ensure they are in degrees.
2. Enter Known Side: Input the length of the side that is opposite Angle A into the ‘Side a’ field.
3. Select Units: Choose the appropriate unit of length for your side from the dropdown menu (e.g., cm, m, inches).
4. Interpret Results: The calculator will instantly display the lengths of the two unknown sides (b and c). It also provides crucial intermediate values like the third angle (Angle C), the triangle’s area, and its perimeter. The visual chart and summary table update in real time.

Key Factors That Affect the Calculation

• Sum of Angles: The two input angles must sum to less than 180 degrees. If they don’t, a valid triangle cannot be formed.
• Input Precision: The accuracy of the calculated side lengths depends directly on the precision of your input values.
• Angle Units: Our calculator uses degrees. If your angles are in radians, you must convert them first.
• Side-Angle Pairing: You must correctly identify which known side is opposite which known angle. Our calculator assumes you know side ‘a’ and angle ‘A’. If you know a different pair (e.g., side ‘b’ and angle ‘B’), you can simply relabel your triangle’s vertices to match the calculator’s inputs.
• Ambiguous Case (SSA): This calculator is designed for AAS and ASA cases. The Side-Side-Angle (SSA) case can sometimes have two possible solutions, one solution, or no solution, and requires a different logical approach which is not covered here. Learn more about it with a triangle solver.
• Rounding: Calculations involving trigonometric functions often result in long decimal numbers. We round the results for clarity, but this can introduce very small differences compared to a manual calculation carried through with full precision.

Frequently Asked Questions (FAQ)

What is the Law of Sines?
The Law of Sines is a formula used in trigonometry that states the ratio of the length of a side of a triangle to the sine of its opposite angle is the same for all three sides.

Can I use this calculator for a right triangle?
Yes. A right triangle is just a special case where one of the angles is 90 degrees. If you know another angle and a side, this calculator will work perfectly.

What if I know two sides and one angle (SSA)?
This calculator is not designed for the SSA case, also known as the ambiguous case. You would typically use the Law of Cosines or a specialized right triangle calculator for that scenario.

Why does the sum of angles have to be less than 180°?
A fundamental property of any triangle in Euclidean geometry is that the sum of its three interior angles is always exactly 180°. If two angles already sum to 180° or more, there is no “room” for a third positive angle, so a triangle cannot be formed.

How do I know if I have an ASA or AAS triangle?
It depends on the position of the known side. If the known side is *between* the two known angles, it’s an ASA (Angle-Side-Angle) triangle. If the known side is *not* between the two known angles, it’s an AAS (Angle-Angle-Side) triangle. This calculator works for both.

What are the units for the Area result?
The area is given in square units corresponding to your selected side length unit (e.g., square cm, square inches).

How is the triangle area calculated?
Once all three sides and angles are known, the area can be calculated using the formula: Area = (1/2) * a * b * sin(C). This calculator uses one of several equivalent formulas.

What happens if my inputs don’t form a valid triangle?
The calculator will display an error message if the sum of the two input angles is 180 degrees or more, as it’s impossible to form a triangle under those conditions.