Inradius Of Equilateral Triangle Given Circumradius Calculator

Formula Used:

\[ Inradius\ of\ Equilateral\ Triangle = \frac{1}{2} \times Circumradius\ of\ Equilateral\ Triangle \] \[ r_i = \frac{1}{2} \times r_c \]

Inradius of Equilateral Triangle:

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1. What is the Inradius of Equilateral Triangle?

The Inradius of an Equilateral Triangle is defined as the radius of the circle which is inscribed inside the triangle, touching all three sides. It represents the distance from the center of the inscribed circle (incenter) to any side of the triangle.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ Inradius\ of\ Equilateral\ Triangle = \frac{1}{2} \times Circumradius\ of\ Equilateral\ Triangle \] \[ r_i = \frac{1}{2} \times r_c \]

Where:

\( r_i \) — Inradius of Equilateral Triangle (m)
\( r_c \) — Circumradius of Equilateral Triangle (m)

Explanation: In an equilateral triangle, the inradius is exactly half the length of the circumradius. This relationship holds true for all equilateral triangles regardless of their size.

3. Importance of Inradius Calculation

Details: Calculating the inradius is important in geometry for determining the area of the inscribed circle, understanding the geometric properties of equilateral triangles, and solving various geometric problems involving circles inscribed in triangles.

4. Using the Calculator

Tips: Enter the circumradius value in meters. The value must be positive and greater than zero. The calculator will compute the corresponding inradius using the formula ri = 1/2 * rc.

5. Frequently Asked Questions (FAQ)

Q1: Why is the inradius exactly half the circumradius in an equilateral triangle?
A: This is a unique geometric property of equilateral triangles where the centroid, circumcenter, and incenter all coincide at the same point, creating this specific ratio.

Q2: Can this formula be used for other types of triangles?
A: No, this specific relationship (ri = 1/2 * rc) only applies to equilateral triangles. Other triangle types have different relationships between inradius and circumradius.

Q3: How is the circumradius related to the side length of an equilateral triangle?
A: The circumradius (rc) of an equilateral triangle with side length 'a' is given by rc = a/√3.

Q4: What are some practical applications of knowing the inradius?
A: Inradius calculations are used in engineering design, architecture, computer graphics, and various geometric optimization problems.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact for perfect equilateral triangles. The accuracy depends on the precision of the input circumradius value.