Find Other Sides Of Triangle Calculator

Easily calculate the unknown sides and angles of a triangle with our Triangle Side Calculator. Enter the known values to get started.

Triangle Calculator

What is a Triangle Side Calculator?

A Triangle Side Calculator is a tool used to determine the unknown lengths of sides and the measures of angles of a triangle when some information about the triangle is already known. Typically, you need at least three pieces of information (like two sides and an angle, or two angles and a side) to define a unique triangle or a limited set of possible triangles. Our Triangle Side Calculator helps solve triangles based on the Side-Angle-Side (SAS) and Angle-Angle-Side (AAS) or Angle-Side-Angle (ASA) congruence/similarity rules.

This calculator is useful for students studying geometry and trigonometry, engineers, architects, and anyone who needs to solve for triangle dimensions. It simplifies the application of the Law of Sines and the Law of Cosines. Common misconceptions are that any three values will define a triangle, but for example, three angles (AAA) only define the shape, not the size, and the sum of any two sides must be greater than the third (Triangle Inequality Theorem).

Triangle Side Calculator Formula and Mathematical Explanation

The Triangle Side Calculator primarily uses two fundamental laws of trigonometry:

• Law of Cosines: Relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides a, b, c, and angles A, B, C opposite to them respectively:
  • c² = a² + b² – 2ab cos(C)
  • a² = b² + c² – 2bc cos(A)
  • b² = a² + c² – 2ac cos(B)

  This is used in the SAS case to find the third side.

• Law of Sines: Relates the lengths of the sides of a triangle to the sines of its angles:
  • a/sin(A) = b/sin(B) = c/sin(C)

  This is used in the AAS/ASA case to find the other sides once all angles are known (the third angle is found using A + B + C = 180°).

For SAS (Side a, Side b, Angle C known):

1. Find side c: c = √(a² + b² – 2ab cos(C))
2. Find Angle A: A = arccos((b² + c² – a²)/(2bc)) (in degrees)
3. Find Angle B: B = 180° – A – C

For AAS/ASA (Angle A, Angle B, Side c known):

1. Find Angle C: C = 180° – A – B
2. Find side a: a = c * sin(A) / sin(C)
3. Find side b: b = c * sin(B) / sin(C)

The area is calculated as 0.5 * a * b * sin(C).

Variable	Meaning	Unit	Typical Range
a, b, c	Lengths of the sides of the triangle	Units (e.g., cm, m, inches)	> 0
A, B, C	Angles of the triangle opposite sides a, b, c respectively	Degrees	> 0 and < 180 (sum = 180)
Area	Area of the triangle	Square Units	> 0

Practical Examples (Real-World Use Cases)

Let’s see how the Triangle Side Calculator works with practical examples.

Example 1 (SAS): Suppose you are surveying a piece of land. You measure two sides to be 100 meters and 120 meters, and the angle between these two sides is 75 degrees.

• Input: Side a = 100, Side b = 120, Angle C = 75°
• Using the calculator (SAS mode):
  • Side c ≈ 131.65 meters
  • Angle A ≈ 47.96°
  • Angle B ≈ 57.04°
  • Area ≈ 5795.55 square meters
• The calculator finds the length of the third side and the other two angles, plus the area.

Example 2 (AAS/ASA): You are at point A and see a tree at point C. You measure the angle between your baseline AB and the tree (Angle A) as 40°. You walk 50 meters to point B and measure the angle between the baseline AB and the tree (Angle B) as 65°.

• Input: Angle A = 40°, Angle B = 65°, Side c (baseline AB) = 50 meters
• Using the calculator (AAS/ASA mode):
  • Angle C = 180 – 40 – 65 = 75°
  • Side a (distance BC) ≈ 33.68 meters
  • Side b (distance AC) ≈ 47.44 meters
  • Area ≈ 798.1 square meters
• The calculator finds the distances from A and B to the tree (sides b and a) and the third angle.

How to Use This Triangle Side Calculator

1. Select Known Information: Choose whether you know “Two Sides and Included Angle (SAS)” or “Two Angles and One Side (AAS/ASA)” using the radio buttons.
2. Enter Values: Input the known side lengths and/or angle measures (in degrees) into the corresponding fields. Ensure angles are between 0 and 180 degrees, and their sum (if two are given) is less than 180.
3. Calculate: The results will update automatically as you type, or you can click the “Calculate” button.
4. View Results: The calculator will display:
   • The calculated side(s) and angle(s) as the primary result.
   • Intermediate values like the third angle (in AAS/ASA) and area.
   • A table summarizing all sides and angles.
   • A bar chart visualizing the side lengths and area.
5. Reset: Click “Reset” to clear the fields and start over with default values.
6. Copy: Click “Copy Results” to copy the main findings to your clipboard.

When making decisions, ensure your input measurements are as accurate as possible, as small errors in input can lead to larger errors in the calculated values, especially with certain triangle configurations.

Key Factors That Affect Triangle Side Calculator Results

The accuracy and validity of the results from the Triangle Side Calculator depend on several factors:

1. Accuracy of Input Values: The precision of the known side lengths and angles directly impacts the calculated results. Small measurement errors can propagate.
2. Chosen Method (SAS, AAS/ASA): You must correctly identify the known parameters to select the right calculation mode.
3. Angle Units: Ensure angles are entered in degrees as the calculator expects.
4. Triangle Inequality Theorem: For any valid triangle, the sum of the lengths of any two sides must be greater than the length of the third side. The calculator implicitly handles this when solving, but you can’t form a triangle with sides 2, 3, and 6, for instance.
5. Sum of Angles: The sum of the interior angles of any planar triangle is always 180 degrees. If you input two angles for AAS/ASA, their sum must be less than 180.
6. Rounding: The number of decimal places used in calculations and displayed results can affect precision. Our calculator uses standard precision.

Frequently Asked Questions (FAQ)

What is the Triangle Inequality Theorem?

It states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side (a + b > c, a + c > b, b + c > a).

Can I use the Triangle Side Calculator for any triangle?

Yes, as long as you have enough information to define it (like SAS or AAS/ASA conditions). It works for scalene, isosceles, and equilateral triangles provided you input the correct initial values.

What if I know three sides (SSS)?

This calculator focuses on finding sides given some angles. To find angles given three sides (SSS), you would primarily use the Law of Cosines to find each angle. Our calculator is currently set up for SAS and AAS/ASA to find sides.

What if I know two sides and a non-included angle (SSA)?

This is the “ambiguous case” and can result in zero, one, or two possible triangles. Our current Triangle Side Calculator doesn’t explicitly handle the SSA case due to its complexity and potential for multiple solutions.

How are the angles measured?

Angles are measured in degrees. Ensure your input is in degrees.

Why does the sum of two angles in AAS/ASA need to be less than 180?

Because the sum of all three angles in a triangle is exactly 180 degrees. If two angles sum to 180 or more, there’s no room for a third positive angle.

Can I calculate the area using this tool?

Yes, the calculator provides the area of the triangle as one of the results.

Is this calculator suitable for right-angled triangles?

Yes, a right-angled triangle is a special case. If you know it’s a right triangle, you can use 90 degrees as one of the angles, or use the Pythagorean theorem if two sides are known (though this calculator uses more general laws).

Related Tools and Internal Resources

Explore more calculators and resources:

• Right Triangle Calculator: Specifically designed for triangles with a 90-degree angle.
• Pythagorean Theorem Calculator: Find the missing side of a right triangle.
• Area Calculator: Calculate the area of various shapes, including triangles with different formulas.
• Angle Converter: Convert between degrees and radians.
• Geometry Formulas: A collection of common geometry formulas.
• Trigonometry Basics: Learn the fundamentals of trigonometry.