Find Sides And Angles Of Triangle Calculator

Easily find missing sides, angles, area, and perimeter of any triangle using our triangle sides and angles calculator.

Triangle Calculator

Property	Value
Side a	–
Side b	–
Side c	–
Angle A	–
Angle B	–
Angle C	–
Area	–
Perimeter	–
Type (Sides)	–
Type (Angles)	–

Summary of triangle properties.

What is a Triangle Sides and Angles Calculator?

A triangle sides and angles calculator is a tool used to determine the unknown lengths of sides, measures of angles, area, and perimeter of a triangle, given certain known properties. Depending on which sides and angles are provided, the calculator employs trigonometric functions like the Law of Sines and the Law of Cosines, as well as the basic principle that the sum of angles in a triangle is 180 degrees, to find the missing values. It's an essential tool for students, engineers, surveyors, and anyone working with geometry or trigonometry.

This calculator can handle various scenarios, including SSS (Side-Side-Side), SAS (Side-Angle-Side), ASA (Angle-Side-Angle), AAS (Angle-Angle-Side), and the ambiguous SSA (Side-Side-Angle) case. By inputting the known values, our triangle sides and angles calculator provides the complete solution for the triangle.

Who should use it? Anyone needing to solve triangles for academic purposes (geometry, trigonometry), professional work (engineering, architecture, surveying, navigation), or even DIY projects involving triangular shapes. Common misconceptions include thinking any three values will define a unique triangle (SSA can yield 0, 1, or 2 triangles) or that only right-angled triangles can be solved (this calculator works for any triangle).

Triangle Sides and Angles Calculator Formulas and Mathematical Explanation

To find the missing sides and angles of a triangle, we use several fundamental laws and principles:

• Sum of Angles: The sum of the interior angles of any triangle is always 180 degrees: A + B + C = 180°.
• Law of Sines: Relates the sides of a triangle to the sines of their opposite angles: a/sin(A) = b/sin(B) = c/sin(C). This is useful for ASA, AAS, and SSA cases.
• Law of Cosines: Relates the length of a side to the lengths of the other two sides and the angle opposite the first side:
  • a² = b² + c² – 2bc cos(A)
  • b² = a² + c² – 2ac cos(B)
  • c² = a² + b² – 2ab cos(C)
  This is particularly useful for SSS (to find angles) and SAS (to find the third side).
• Area Formula: Area = 0.5 * a * b * sin(C), or using Heron's formula for SSS where s = (a+b+c)/2 is the semi-perimeter: Area = sqrt(s(s-a)(s-b)(s-c)).

The triangle sides and angles calculator first identifies the given case (SSS, SAS, ASA, AAS, or SSA) and then applies the appropriate formulas.

Variables Table

Variable	Meaning	Unit	Typical Range
a, b, c	Lengths of the sides of the triangle	Length units (e.g., m, cm, inches)	> 0
A, B, C	Measures of the interior angles opposite sides a, b, c respectively	Degrees	0° – 180°
Area	The space enclosed by the triangle	Square length units	> 0
Perimeter	The sum of the lengths of the sides (a+b+c)	Length units	> 0

Variables used in the triangle sides and angles calculator.

Practical Examples (Real-World Use Cases)

Example 1: SSS (Side-Side-Side)

Suppose you have a triangular piece of land with sides a = 7 meters, b = 10 meters, and c = 12 meters. Using the Law of Cosines:

cos(A) = (10² + 12² – 7²) / (2 * 10 * 12) ≈ 0.8125 => A ≈ 35.66°

cos(B) = (7² + 12² – 10²) / (2 * 7 * 12) ≈ 0.5536 => B ≈ 56.38°

C = 180° – 35.66° – 56.38° ≈ 87.96°

The triangle sides and angles calculator would confirm these angles and also calculate the area.

Example 2: SAS (Side-Angle-Side)

Imagine you know two sides of a roof truss are a = 5 ft and c = 8 ft, and the included angle B = 60°. To find the third side b:

b² = 5² + 8² – 2 * 5 * 8 * cos(60°) = 25 + 64 – 80 * 0.5 = 49 => b = 7 ft

Then, using the Law of Sines, you can find angles A and C. Our triangle sides and angles calculator does this automatically.

For more examples, check our triangle area calculator or Pythagorean theorem calculator for right triangles.

How to Use This Triangle Sides and Angles Calculator

1. Select the Given Case: Choose the combination of sides and angles you know from the "Given Information" dropdown (SSS, SAS, ASA, AAS, or SSA).
2. Enter Known Values: Input the values for the sides and angles corresponding to your selected case. Only the relevant input fields will be enabled. Ensure angles are in degrees.
3. Click Calculate: The calculator will automatically update as you type, or you can click the "Calculate" button.
4. Review Results: The calculator will display the unknown sides, angles, area, and perimeter in the results section, along with a summary table and a chart. For the SSA case, it will indicate if there are 0, 1, or 2 solutions.
5. Reset or Copy: Use the "Reset" button to clear inputs and start over, or "Copy Results" to copy the findings.

Understanding the results helps in various applications, from checking homework to designing structures.

Key Factors That Affect Triangle Sides and Angles Calculator Results

• Given Information: The combination of sides and angles you start with (SSS, SAS, etc.) dictates the solution method and the uniqueness of the solution (especially with SSA).
• Accuracy of Input Values: Small errors in input lengths or angles can lead to larger inaccuracies in the calculated values, especially when using trigonometric functions.
• The Ambiguous Case (SSA): When given two sides and a non-included angle, there might be zero, one, or two possible triangles. Our triangle sides and angles calculator will address this.
• Triangle Inequality Theorem: For the SSS case, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. If not, no triangle can be formed.
• Angle Sum: The sum of angles must be 180°. If given two angles, the third is fixed, which is used in ASA and AAS cases.
• Units: Ensure all side lengths are in the same units for consistent area and perimeter calculations. Angles are always in degrees for this calculator.

Our geometry calculators cover various shapes and their properties.

Frequently Asked Questions (FAQ)

1. Can I solve a triangle if I only know two angles?

No, knowing only two angles (which means you know all three because they sum to 180°) defines the shape (similarity) but not the size. You need at least one side length (ASA or AAS) to find the other sides.

2. What is the ambiguous case (SSA)?

The SSA case occurs when you know two sides and a non-included angle. Depending on the values, there can be 0, 1, or 2 possible triangles that fit the criteria. Our triangle sides and angles calculator attempts to find all valid solutions.

3. How is the area of the triangle calculated?

If two sides and the included angle are known (e.g., b, c, A), Area = 0.5 * b * c * sin(A). If all three sides are known (SSS), Heron's formula is used. The calculator selects the appropriate formula.

4. What if the SSS inputs don't form a triangle?

If the sum of two sides is not greater than the third side (e.g., a+b ≤ c), the calculator will indicate that no valid triangle can be formed with those side lengths.

5. Does this calculator work for right-angled triangles?

Yes, it works for any triangle, including right-angled ones. You can also use our specific right triangle calculator.

6. Why are my angle results slightly different from other calculators?

This might be due to rounding during calculations or the precision of trigonometric function implementations. Our triangle sides and angles calculator aims for high precision.

7. What units should I use for sides?

You can use any consistent unit of length (meters, feet, inches, etc.). The area will be in square units and perimeter in the same units.

8. How do I interpret the chart?

The chart provides a visual comparison of the calculated side lengths and angles, helping you understand the proportions of the triangle.