Inradius of Equilateral Triangle given Exradius Calculator

Formula Used:

\[ \text{Inradius of Equilateral Triangle} = \frac{1}{3} \times \text{Exradius of Equilateral Triangle} \] \[ r_i = \frac{1}{3} \times r_e \]

Inradius of Equilateral Triangle:

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1. What is the Inradius of Equilateral Triangle?

The Inradius of Equilateral Triangle is defined as the radius of the circle which is inscribed inside the triangle. It's the distance from the center of the inscribed circle (incircle) to any side of the equilateral triangle.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ r_i = \frac{1}{3} \times r_e \]

Where:

\( r_i \) — Inradius of Equilateral Triangle (m)
\( r_e \) — Exradius of Equilateral Triangle (m)

Explanation: In an equilateral triangle, the inradius is exactly one-third of the exradius. This relationship holds true for all equilateral triangles due to their perfect symmetry.

3. Importance of Inradius Calculation

Details: Calculating the inradius is important in geometry for determining the size of the largest circle that can fit inside an equilateral triangle. This has applications in various fields including engineering, architecture, and design where circular components need to fit within triangular spaces.

4. Using the Calculator

Tips: Enter the exradius value in meters. The value must be positive and greater than zero. The calculator will compute the corresponding inradius.

5. Frequently Asked Questions (FAQ)

Q1: What is the relationship between inradius and exradius in an equilateral triangle?
A: In an equilateral triangle, the inradius is exactly one-third of the exradius.

Q2: Can this formula be used for other types of triangles?
A: No, this specific relationship (ri = re/3) only applies to equilateral triangles due to their perfect symmetry.

Q3: What are typical units for measuring inradius?
A: Inradius can be measured in any unit of length (meters, centimeters, inches, etc.), but consistency with the exradius unit is important.

Q4: How does the inradius relate to the side length of an equilateral triangle?
A: The inradius can also be calculated as \( r_i = \frac{a\sqrt{3}}{6} \), where a is the side length.

Q5: What is the geometric significance of the inradius?
A: The inradius represents the distance from the triangle's center to any side, and it's the radius of the largest circle that can fit inside the triangle.