Triangle Side Calculator: Find Sides with Angles


Triangle Side & Angle Calculator

This powerful tool helps with the complex task of calculating sides of a triangle using angle and side information. Whether you’re a student, engineer, or hobbyist, you can solve for unknown triangle properties using trigonometric laws like the Law of Sines and the Law of Cosines. Simply input the values you know to get started.



In degrees (°)


Opposite Angle A


In degrees (°)


Opposite Angle B


In degrees (°)


Opposite Angle C



What is Calculating Sides of a Triangle Using Angle and Side?

“Calculating sides of a triangle using angle and side” is the process of determining the unknown dimensions (lengths of sides and measures of angles) of a triangle when you have a partial set of its measurements. This is a fundamental concept in trigonometry and geometry, essential for fields like engineering, architecture, physics, and even video game design. You can often find a missing side with a right triangle calculator, but for non-right triangles, more advanced rules are needed.

The process relies on established mathematical principles, primarily the Law of Sines and the Law of Cosines. These laws provide relationships between the sides and angles of any triangle (not just right-angled ones), allowing for a complete “solution” to be found as long as a minimum amount of information is known. Typically, you need at least three pieces of information, including at least one side length, to solve the triangle.

Formulas for Calculating Sides and Angles

To solve a triangle, we use two primary formulas. The choice of formula depends on the information you are given.

The Law of Sines

The Law of Sines states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant for all three sides. It’s most useful for cases where you know two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA).

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

The Law of Cosines

The Law of Cosines relates the length of one side to the lengths of the other two sides and the angle between them. It is used when you know two sides and the included angle (SAS) or all three sides (SSS).

Formula: c² = a² + b² - 2ab cos(C)

Variable Explanations
Variable Meaning Unit Typical Range
a, b, c Length of the sides of the triangle cm, m, in, ft, etc. Any positive number
A, B, C Interior angles opposite sides a, b, and c Degrees (°) 0° – 180°

Practical Examples

Example 1: Given Two Angles and a Side (AAS)

Imagine you are surveying a piece of land. You know Angle A = 45°, Angle B = 60°, and the side opposite Angle A (Side a) is 100 meters.

  • Inputs: Angle A = 45°, Angle B = 60°, Side a = 100 m
  • Step 1 (Find Angle C): Angle C = 180° – 45° – 60° = 75°
  • Step 2 (Find Side b using Law of Sines): b = (a * sin(B)) / sin(A) = (100 * sin(60°)) / sin(45°) ≈ 122.47 m
  • Step 3 (Find Side c using Law of Sines): c = (a * sin(C)) / sin(A) = (100 * sin(75°)) / sin(45°) ≈ 136.60 m
  • Results: Side b ≈ 122.47 m, Side c ≈ 136.60 m, Angle C = 75°

Example 2: Given Two Sides and an Included Angle (SAS)

An architect is designing a triangular roof section. They know Side a = 10 ft, Side b = 12 ft, and the included angle between them (Angle C) is 30°.

  • Inputs: Side a = 10 ft, Side b = 12 ft, Angle C = 30°
  • Step 1 (Find Side c using Law of Cosines): c² = 10² + 12² – 2(10)(12)cos(30°) = 100 + 144 – 240(√3/2) ≈ 36.15. So, c ≈ √36.15 ≈ 6.01 ft.
  • Step 2 (Find Angle A using Law of Sines): sin(A)/a = sin(C)/c => sin(A) = (10 * sin(30°)) / 6.01 ≈ 0.832. So, A = arcsin(0.832) ≈ 56.3°
  • Step 3 (Find Angle B): Angle B = 180° – 30° – 56.3° = 93.7°
  • Results: Side c ≈ 6.01 ft, Angle A ≈ 56.3°, Angle B ≈ 93.7°. This problem could be solved easily with a law of cosines calculator.

How to Use This Triangle Side Calculator

Using our tool for calculating sides of a triangle using angle and side is straightforward. Follow these steps for accurate results:

  1. Enter Known Values: Fill in the input fields for the angles and/or sides you know. You need at least three values, including at least one side.
  2. Select Units: Choose the appropriate unit of measurement for your side lengths from the dropdown menu.
  3. Calculate: The calculator will automatically update as you type. You can also press the “Calculate” button.
  4. Review Results: The results section will display all three sides, all three angles, the perimeter, and the area of the triangle.
  5. Visualize: A dynamic SVG chart will draw the solved triangle to scale, providing a helpful visual representation. Using a triangle solver helps to visualize these relationships.

Key Factors That Affect Calculations

  • Input Accuracy: The precision of your results depends entirely on the accuracy of your initial measurements. Small errors in angles or lengths can lead to larger discrepancies in the calculated values.
  • Correct Formula Application: The calculator automatically determines whether to use the Law of Sines or Law of Cosines based on your inputs. Manual calculations require choosing the correct formula for the given scenario (e.g., SAS vs. ASA).
  • Unit Consistency: Always ensure all side lengths are in the same unit. Our calculator handles this with a single unit selector, but it’s a critical factor in manual calculations.
  • The Ambiguous Case (SSA): When given two sides and a non-included angle (Side-Side-Angle), there can sometimes be two possible triangles, one triangle, or no triangle. Our calculator attempts to find a valid solution, but this is a known complexity in trigonometry.
  • Rounding: For maximum accuracy, it’s best to use unrounded numbers throughout the entire calculation process and only round the final answer. Our trigonometry calculator does this automatically.
  • Angle Sum Property: The three interior angles of any valid Euclidean triangle must add up to exactly 180°. If your known angles already exceed this, a triangle cannot be formed.

Frequently Asked Questions (FAQ)

Can I use this for a right-angled triangle?

Yes, this calculator works for any triangle, including right-angled ones. Simply enter 90 degrees for one of the angles. However, for more specific features, a dedicated Pythagorean theorem calculator might be more efficient.

What does “NaN” in the result mean?

“NaN” stands for “Not a Number.” This result appears if the input values are invalid or do not form a real triangle (e.g., angles summing to over 180°, or side lengths that violate the triangle inequality theorem).

Why do I need at least one side length?

If you only know the angles, you define the shape of the triangle, but not its size. There are infinitely many triangles (of different sizes) with the same three angles. A side length is needed to set the scale.

What is the Law of Sines?

It’s a formula that states the ratio of a side’s length to the sine of its opposite angle is the same for all three sides of a triangle. It’s crucial for solving triangles when you have an angle-side pair.

What is the Law of Cosines?

It’s a formula that relates the three sides of a triangle to the cosine of one of its angles. It’s a generalization of the Pythagorean theorem and is essential for solving SAS and SSS triangles.

What units can I use?

Our calculator supports centimeters, meters, inches, and feet. You can select your preferred unit, and all calculations will be adjusted accordingly.

How is the area calculated?

Once two sides and the included angle are known (e.g., sides ‘a’ and ‘b’, and angle ‘C’), the area is calculated using the formula: Area = 0.5 * a * b * sin(C).

What is the relationship between sides and angles?

In any triangle, the longest side is opposite the largest angle, and the shortest side is opposite the shortest angle. This fundamental rule, explored with a angle calculator, governs the geometry of all triangles.

Related Tools and Internal Resources

Explore other calculators and resources to deepen your understanding of geometry and trigonometry:

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