Find the Length of the Third Side Triangle Calculator
This calculator helps you find the length of the third side of a triangle when you know the lengths of the other two sides and the angle between them, using the Law of Cosines. You can also use it for right-angled triangles.
Area of Triangle: —
Perimeter of Triangle: —
Angles: A=–, B=–, C=– degrees
c² = a² + b² - 2ab * cos(C)Summary Table
| Parameter | Value |
|---|---|
| Side a | 10 |
| Side b | 15 |
| Angle C (degrees) | 60 |
| Side c | — |
| Area | — |
| Perimeter | — |
Table showing the input values and calculated results for the triangle.
Triangle Visualization
Visual representation of the triangle based on inputs (not perfectly to scale).
What is a Find the Length of the Third Side Triangle Calculator?
A find the length of the third side triangle calculator is a tool used to determine the length of one side of a triangle when the lengths of the other two sides and the angle between them are known. It primarily uses the Law of Cosines for this calculation. It can also be implicitly used for right-angled triangles (where the angle is 90 degrees), in which case the Law of Cosines simplifies to the Pythagorean theorem.
This calculator is useful for students, engineers, architects, and anyone dealing with geometric problems involving triangles. It saves time and reduces the chance of manual calculation errors. A reliable find the length of the third side triangle calculator is essential for accurate geometric analysis.
Common misconceptions include thinking it only works for right-angled triangles or that you need all angles. The most common version of the find the length of the third side triangle calculator (using the Law of Cosines) works for any triangle, provided you have two sides and the included angle.
Find the Length of the Third Side Triangle Calculator Formula and Mathematical Explanation
The primary formula used by the find the length of the third side triangle calculator when two sides and the included angle are known is the Law of Cosines:
c² = a² + b² - 2ab * cos(C)
Where:
- ‘a’ and ‘b’ are the lengths of the two known sides.
- ‘C’ is the angle between sides ‘a’ and ‘b’.
- ‘c’ is the length of the unknown side opposite angle ‘C’.
To find ‘c’, we take the square root:
c = √(a² + b² - 2ab * cos(C))
The angle ‘C’ must be converted from degrees to radians for the `cos` function in JavaScript (`radians = degrees * Math.PI / 180`).
If angle ‘C’ is 90 degrees (a right-angled triangle), cos(90°) = 0, and the formula simplifies to the Pythagorean theorem: c² = a² + b².
The calculator can also find the area using: Area = 0.5 * a * b * sin(C), and other angles using the Law of Sines once ‘c’ is known.
Variables Table
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| a | Length of side ‘a’ | Length units (e.g., cm, m, inches) | > 0 |
| b | Length of side ‘b’ | Length units (e.g., cm, m, inches) | > 0 |
| C | Angle between sides ‘a’ and ‘b’ | Degrees | 0 < C < 180 |
| c | Length of side ‘c’ (opposite angle C) | Length units (e.g., cm, m, inches) | > 0 |
| Area | Area of the triangle | Square length units | > 0 |
Practical Examples
Example 1: Oblique Triangle
Suppose you have a triangle with side a = 7 cm, side b = 10 cm, and the angle C between them is 45 degrees.
- a = 7
- b = 10
- C = 45°
Using the find the length of the third side triangle calculator (Law of Cosines):
c² = 7² + 10² - 2 * 7 * 10 * cos(45°)
c² = 49 + 100 - 140 * 0.7071 ≈ 149 - 98.994 ≈ 50.006
c ≈ √50.006 ≈ 7.07 cm
The length of the third side ‘c’ is approximately 7.07 cm.
Example 2: Right-Angled Triangle
Suppose side a = 3 units, side b = 4 units, and the angle C between them is 90 degrees.
- a = 3
- b = 4
- C = 90°
Using the find the length of the third side triangle calculator:
c² = 3² + 4² - 2 * 3 * 4 * cos(90°)
c² = 9 + 16 - 24 * 0 = 25
c = √25 = 5 units
This matches the Pythagorean theorem (3-4-5 triangle).
How to Use This Find the Length of the Third Side Triangle Calculator
- Enter Side ‘a’: Input the length of the first known side into the “Length of Side ‘a'” field.
- Enter Side ‘b’: Input the length of the second known side into the “Length of Side ‘b'” field.
- Enter Angle ‘C’: Input the angle (in degrees) between sides ‘a’ and ‘b’ into the “Angle ‘C'” field. Make sure the angle is greater than 0 and less than 180. For right triangles, enter 90.
- Calculate: The calculator will automatically update the results as you type. You can also click the “Calculate Side ‘c'” button.
- Read Results: The primary result is the length of side ‘c’. Intermediate results like Area, Perimeter, and other angles (A and B, calculated using the Law of Sines after ‘c’ is found) are also displayed.
- Reset: Click “Reset” to clear the fields to default values.
- Copy: Click “Copy Results” to copy the main findings.
The visualization and table provide a quick summary of your triangle’s properties. Our geometry formulas page has more details.
Key Factors That Affect Triangle Side Length Results
- Lengths of Known Sides (a and b): The longer the known sides, generally, the longer the third side can be, depending on the angle.
- Included Angle (C): This is crucial. As angle C increases towards 180 degrees, side ‘c’ increases. As it decreases towards 0, ‘c’ decreases. At 90 degrees, it aligns with Pythagoras.
- Units of Measurement: Ensure both side ‘a’ and side ‘b’ are in the same units. The result ‘c’ will be in those same units.
- Accuracy of Input: Small errors in the input values, especially the angle, can lead to noticeable differences in the calculated length of side ‘c’.
- Triangle Inequality Theorem: For a valid triangle, the sum of the lengths of any two sides must be greater than the length of the third side (a+b > c, a+c > b, b+c > a). The calculator implicitly respects this based on the Law of Cosines inputs. If you try to form a triangle with an angle too close to 0 or 180 with very different side lengths, ‘c’ might approach |a-b| or a+b.
- Angle Range: The included angle ‘C’ must be between 0 and 180 degrees (exclusive) for a valid triangle. Our find the length of the third side triangle calculator validates this.
Frequently Asked Questions (FAQ)
A: The Law of Cosines is a formula relating the lengths of the sides of a triangle to the cosine of one of its angles: c² = a² + b² – 2ab * cos(C). It’s used in our find the length of the third side triangle calculator.
A: Yes. If you enter 90 degrees for Angle ‘C’, the find the length of the third side triangle calculator effectively uses the Pythagorean theorem (c² = a² + b²) because cos(90°) = 0. See our right triangle calculator for more.
A: This specific find the length of the third side triangle calculator is for two sides and the included angle. For other scenarios, you’d use the Law of Sines or other triangle properties. You might need a more general triangle solver.
A: You can use any units (cm, meters, inches, feet), but be consistent for both side ‘a’ and ‘b’. The result ‘c’ will be in the same units.
A: Once side ‘c’ is found, angles A and B can be found using the Law of Sines (a/sin(A) = b/sin(B) = c/sin(C)) or the Law of Cosines rearranged to find angles.
A: The calculator restricts the angle to be between 0 and 180 degrees (exclusive) because at 0 or 180 degrees, the three points form a straight line, not a triangle.
A: It’s vital in fields like engineering, physics, navigation, and construction for calculating distances and dimensions where direct measurement is difficult but side lengths and angles are known or can be measured.
A: The area is calculated using the formula: Area = 0.5 * a * b * sin(C), where C is the included angle in radians. Check our area of triangle calculator.
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