Breadth of Rectangle given Perimeter and Circumradius Calculator

Formula Used:

\[ Breadth\ of\ Rectangle = \frac{1}{4} \times \left( Perimeter\ of\ Rectangle - \sqrt{8 \times (2 \times Circumradius\ of\ Rectangle)^2 - Perimeter\ of\ Rectangle^2} \right) \]

Breadth of Rectangle:

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1. What is the Breadth of Rectangle Formula?

The formula calculates the breadth of a rectangle when given its perimeter and circumradius. It provides an alternative method to find the shorter side of a rectangle using these specific geometric properties.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ b = \frac{1}{4} \times \left( P - \sqrt{8 \times (2r_c)^2 - P^2} \right) \]

Where:

\( b \) — Breadth of Rectangle (m)
\( P \) — Perimeter of Rectangle (m)
\( r_c \) — Circumradius of Rectangle (m)

Explanation: The formula derives from the relationship between a rectangle's perimeter, circumradius, and its side lengths, using the Pythagorean theorem and algebraic manipulation.

3. Importance of Breadth Calculation

Details: Calculating the breadth of a rectangle is essential in various geometric applications, construction projects, and design calculations where specific dimensional relationships must be maintained.

4. Using the Calculator

Tips: Enter the perimeter and circumradius values in meters. Both values must be positive numbers. The circumradius must be large enough to satisfy the geometric constraints of a rectangle.

5. Frequently Asked Questions (FAQ)

Q1: What is the circumradius of a rectangle?
A: The circumradius is the radius of the circle that passes through all four vertices of the rectangle. It's equal to half the length of the rectangle's diagonal.

Q2: Can this formula be used for squares?
A: Yes, for squares (where length equals breadth), the formula will work correctly as squares are a special case of rectangles.

Q3: What if I get a negative value under the square root?
A: This indicates that the input values are geometrically impossible for a rectangle. The circumradius must be at least half the perimeter divided by 2√2 for a valid rectangle.

Q4: How accurate is this calculation?
A: The calculation is mathematically exact, provided the input values satisfy the geometric constraints of a rectangle.

Q5: Can I use different units of measurement?
A: Yes, as long as both inputs use the same units, the result will be in those same units.