Perimeter Of Equilateral Triangle Given Exradius Calculator

Formula Used:

\[ Perimeter\ of\ Equilateral\ Triangle = 2 \times \sqrt{3} \times Exradius\ of\ Equilateral\ Triangle \] \[ P = 2 \times \sqrt{3} \times r_e \]

Perimeter of Equilateral Triangle:

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1. What is the Perimeter of Equilateral Triangle given Exradius?

The perimeter of an equilateral triangle can be calculated using the exradius (radius of the excircle) through the mathematical relationship between these two geometric properties. This formula provides an alternative method to calculate the perimeter when the exradius is known.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ Perimeter\ of\ Equilateral\ Triangle = 2 \times \sqrt{3} \times Exradius\ of\ Equilateral\ Triangle \] \[ P = 2 \times \sqrt{3} \times r_e \]

Where:

\( P \) — Perimeter of Equilateral Triangle (m)
\( r_e \) — Exradius of Equilateral Triangle (m)
\( \sqrt{3} \) — Square root of 3 (approximately 1.732)

Explanation: This formula establishes the direct proportional relationship between the perimeter of an equilateral triangle and its exradius, with the constant of proportionality being \( 2\sqrt{3} \).

3. Importance of Perimeter Calculation

Details: Calculating the perimeter of geometric shapes is fundamental in various fields including architecture, engineering, and mathematics. For equilateral triangles, knowing the perimeter helps in determining material requirements, boundary measurements, and understanding geometric properties.

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 perimeter of the equilateral triangle.

5. Frequently Asked Questions (FAQ)

Q1: What is an exradius in an equilateral triangle?
A: The exradius is the radius of an excircle (escribed circle) of the triangle, which is a circle tangent to one side of the triangle and the extensions of the other two sides.

Q2: How is this formula derived?
A: The formula is derived from the geometric relationships between the sides, angles, and various radii (inradius, circumradius, exradius) of an equilateral triangle.

Q3: Can this formula be used for other types of triangles?
A: No, this specific formula applies only to equilateral triangles where all sides and angles are equal.

Q4: What are the units of measurement?
A: Both perimeter and exradius are typically measured in meters (m), but any consistent unit of length can be used.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact, provided the input value is accurate and the triangle is perfectly equilateral.