Find The Length Of A Triangle With Angles Calculator

Easily find the unknown sides of a triangle using our triangle side length calculator when you know two angles and one side (opposite one of the angles).

Calculate Triangle Sides (Law of Sines)

Parameter	Value
Angle A
Angle B
Angle C
Side a
Side b
Side c
Area
Perimeter

Summary of triangle properties based on your inputs.

What is a triangle side length calculator?

A triangle side length calculator is a tool used to determine the lengths of the unknown sides of a triangle when certain other properties, such as angles and at least one side length, are known. Specifically, when you know two angles and the side opposite one of them, or two sides and the angle between them, you can find the remaining sides and angles. Our calculator primarily uses the Law of Sines, which is ideal when you have information about two angles and one side (AAS or ASA cases in triangle congruence).

Anyone studying or working with geometry, trigonometry, engineering, architecture, or even fields like navigation and astronomy can benefit from a triangle side length calculator. It simplifies complex calculations and provides quick results. Common misconceptions include thinking any three values will define a triangle (e.g., three angles don’t define side lengths, only shape) or that only right-angled triangles can be solved (the Law of Sines and Cosines work for any triangle).

Triangle Side Length Formula (Law of Sines) and Mathematical Explanation

The primary formula used by this triangle side length calculator when given two angles and a side is the Law of Sines. It states that for any triangle with angles A, B, C and sides opposite to them a, b, c respectively:

a / sin(A) = b / sin(B) = c / sin(C) = 2R (where R is the circumradius)

If we know angle A, angle B, and side a, we can find angle C first:

C = 180° – A – B

Then we can find side b and side c using the Law of Sines:

b = (a * sin(B)) / sin(A)

c = (a * sin(C)) / sin(A)

The area of the triangle can also be calculated as: Area = 0.5 * a * b * sin(C).

Variable	Meaning	Unit	Typical Range
A, B, C	Angles of the triangle	Degrees	> 0, < 180
a, b, c	Sides opposite angles A, B, C	Length units (e.g., m, cm, inches)	> 0
sin(A), sin(B), sin(C)	Sine of the angles	Dimensionless	0 to 1 (for angles 0-180)

Variables used in the Law of Sines.

Practical Examples (Real-World Use Cases)

Example 1: Surveying Land

A surveyor wants to find the distance across a river (side ‘b’). They stand at point A, measure the angle to a tree on the other side (point B) relative to a baseline along their side of the river to point C. Let’s say they measure angle A = 60°, angle C = 50°, and the baseline AC (side b) is 100 meters. Wait, this setup needs Law of Cosines if AC is known. Let’s rephrase for Law of Sines.

They stand at point C and sight point A and point B across the river. They measure angle C = 50°. They move to point A along the river bank, a distance of 100m (side b = 100m), and measure angle A = 60°. They want to find the distance AB (side c). First find angle B = 180 – 60 – 50 = 70°. Using Law of sines c/sin(C) = b/sin(B) is not what we have. We have A, C and b, need c. Let’s assume we know A, B and a.

Surveyor at point A measures angle to point B across river as 65° (Angle A). Walks 100m to point C, measures angle to B as 55° (Angle C). Then angle B is 180-65-55 = 60°. Side AC (b)=100m. To find AB (c): c/sin(55) = 100/sin(60) => c = 100 * sin(55)/sin(60) approx 94.5m. Let’s use our calculator’s input structure: If Angle A=65, Angle B=60, Side a=? No, we need one side opposite one of these. If Angle A=65, Side a=94.5, Angle C=55. Angle B=60. Side b = 94.5 * sin(60)/sin(65) = 90.3m. Side c = 94.5 * sin(55)/sin(65) = 85.3m

So, input Angle A=65°, Angle B=60°, Side a=94.5m. Result: Angle C=55°, Side b≈90.3m, Side c≈85.3m.

Example 2: Navigation

A ship at sea observes a lighthouse at a bearing corresponding to an angle of 35° (let’s call this our angle A relative to some baseline). The ship travels 10 nautical miles along its course, and then observes the lighthouse at an angle of 60° (now part of angle B, or contributing to it). If the ship’s course and the lighthouse form a triangle, and we know the angle the ship’s course makes with the initial lighthouse observation, we could use the triangle side length calculator. Suppose initial angle at ship (A) to lighthouse is 35°, ship travels 10 miles (side c), new angle at ship (B) is 60°. No, this isn’t directly AAS.
Let’s simplify: Ship at A, angle to lighthouse C is 35°. Ship travels 10 miles to B, angle to lighthouse C is now 50°. So, angle CAB = 35°, angle CBA = 180-50 = 130°, side AB (c) = 10. Angle ACB = 180-35-130 = 15°. We know c, A, B. Need a or b. Our calculator needs two angles and one side *opposite*.
Okay, simpler: Angle A = 40°, Angle B = 70°, Side a = 15 km. The triangle side length calculator finds Angle C = 180 – 40 – 70 = 70°. Side b = (15 * sin(70))/sin(40) ≈ 21.9 km, Side c = (15 * sin(70))/sin(40) ≈ 21.9 km (Isosceles triangle).

How to Use This triangle side length calculator

1. Enter Angle A: Input the value of the first known angle (in degrees) into the “Angle A” field.
2. Enter Angle B: Input the value of the second known angle (in degrees) into the “Angle B” field. The sum of A and B must be less than 180.
3. Enter Side a: Input the length of the side opposite Angle A into the “Side a” field.
4. View Results: The calculator automatically calculates Angle C, Side b, Side c, the area, and the perimeter of the triangle. The results are displayed in the “Results” section, the summary table, and visualized in the chart.
5. Interpret: Side b and Side c are the lengths you were looking for. Angle C is the third angle. The table and chart give a good overview.
6. Reset or Copy: Use the “Reset” button to clear inputs to defaults or “Copy Results” to copy the main findings.

Use the triangle side length calculator whenever you have an AAS or ASA scenario in trigonometry.

Key Factors That Affect triangle side length calculator Results

• Accuracy of Angle Measurements: Small errors in angle measurements can lead to significant differences in calculated side lengths, especially when angles are very small or close to 180°.
• Accuracy of Known Side Length: The precision of the input side ‘a’ directly affects the precision of the calculated sides ‘b’ and ‘c’.
• Sum of Angles A and B: If A + B is close to 180°, angle C will be very small, and the triangle will be very thin, potentially leading to large side lengths ‘b’ and ‘c’ if ‘a’ is not correspondingly small. The sine of a small angle is small, which can amplify errors.
• Units Used: Ensure consistent units are used for side ‘a’. The output units for ‘b’ and ‘c’ will be the same as the input units for ‘a’.
• Rounding: The number of decimal places used in calculations and displayed results can affect precision. Our triangle side length calculator uses standard JavaScript precision.
• Triangle Inequality: Although calculated here, remember that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. The Law of Sines will always produce sides that satisfy this if the angles are valid.

Frequently Asked Questions (FAQ)

What is the Law of Sines?

The Law of Sines is a formula relating the lengths of the sides of any triangle to the sines of its angles: a/sin(A) = b/sin(B) = c/sin(C).

When can I use this triangle side length calculator?

You can use it when you know two angles and one side of a triangle (AAS or ASA cases). If you know two sides and the included angle, or three sides, you’d use the Law of Cosines, which you can find in our law of sines calculator (which often covers cosines too) or a dedicated Law of Cosines tool.

What if the sum of my angles A and B is 180° or more?

The calculator will show an error or produce invalid results because the three angles of a triangle must sum to exactly 180°.

Can I find the angles if I know the sides?

Yes, but you would use the Law of Cosines for that, or our triangle angle calculator.

Does this calculator work for right-angled triangles?

Yes, it works for any triangle, including right-angled ones, although for right triangles, basic sine, cosine, and tangent ratios or the Pythagorean theorem are often simpler. Check our right triangle calculator.

What units should I use for the side length?

You can use any unit of length (meters, feet, inches, etc.) for side ‘a’, and the results for sides ‘b’ and ‘c’ will be in the same unit.

How accurate are the results from the triangle side length calculator?

The results are as accurate as the input values and the precision of the sine function in JavaScript. For most practical purposes, the accuracy is very high.

Where can I learn more about trigonometry?

You can explore resources on trigonometry, including our trigonometry solver and other geometry calculators.