Shorter Side of Scalene Triangle given Smaller Angle and other Sides Calculator

Formula Used:

\[ \text{Shorter Side} = \sqrt{\text{Longer Side}^2 + \text{Medium Side}^2 - 2 \times \text{Longer Side} \times \text{Medium Side} \times \cos(\text{Smaller Angle})} \]

Longer Side of Scalene Triangle:

Medium Side of Scalene Triangle:

Smaller Angle of Scalene Triangle:

Shorter Side of Scalene Triangle:

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1. What is the Shorter Side of Scalene Triangle Formula?

The formula calculates the length of the shorter side of a scalene triangle when the longer side, medium side, and the smaller angle are known. This is derived from the Law of Cosines, which relates the lengths of the sides of a triangle to the cosine of one of its angles.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \text{Shorter Side} = \sqrt{\text{Longer Side}^2 + \text{Medium Side}^2 - 2 \times \text{Longer Side} \times \text{Medium Side} \times \cos(\text{Smaller Angle})} \]

Where:

Longer Side — The longest side of the scalene triangle (m)
Medium Side — The second longest side of the scalene triangle (m)
Smaller Angle — The angle opposite to the shorter side (degrees)
cos — Cosine trigonometric function
sqrt — Square root function

Explanation: The formula applies the Law of Cosines to find the unknown side length when two sides and the included angle are known.

3. Importance of Calculating the Shorter Side

Details: Calculating the shorter side is essential for complete triangle analysis, construction planning, and various engineering applications where all side lengths must be determined for structural integrity.

4. Using the Calculator

Tips: Enter the longer side and medium side in meters, and the smaller angle in degrees (between 0 and 180). All values must be positive numbers.

5. Frequently Asked Questions (FAQ)

Q1: Why use the Law of Cosines instead of the Law of Sines?
A: The Law of Cosines is used when two sides and the included angle are known, while the Law of Sines is typically used when two angles and one side are known.

Q2: What if the angle is not the smaller angle?
A: The formula specifically requires the smaller angle (angle opposite the shorter side) to calculate the correct shorter side length.

Q3: Can this formula be used for any triangle?
A: Yes, the Law of Cosines applies to all triangles, but this specific formulation is optimized for finding the shorter side in scalene triangles.

Q4: What are the limitations of this calculation?
A: The calculation assumes accurate input measurements and that the triangle inequality theorem is satisfied (sum of any two sides greater than the third side).

Q5: How precise should the angle measurement be?
A: For accurate results, the angle should be measured as precisely as possible, as small angle errors can significantly affect the calculated side length.