Circumradius of Cyclic Quadrilateral Given Perimeter Calculator

Circumradius of Cyclic Quadrilateral Formula:

\[ R = \frac{1}{4} \times \sqrt{\frac{((ab + cd)(ac + bd)(ad + bc))}{(s-a)(s-b)(s-c)(s-d)}} \]

Side A of Cyclic Quadrilateral:

Side B of Cyclic Quadrilateral:

Side C of Cyclic Quadrilateral:

Side D of Cyclic Quadrilateral:

Perimeter of Cyclic Quadrilateral:

Circumradius of Cyclic Quadrilateral:

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1. What is the Circumradius of a Cyclic Quadrilateral?

The circumradius of a cyclic quadrilateral is the radius of the circumscribed circle that passes through all four vertices of the quadrilateral. A cyclic quadrilateral is a four-sided polygon where all vertices lie on a single circle.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ R = \frac{1}{4} \times \sqrt{\frac{((ab + cd)(ac + bd)(ad + bc))}{(s-a)(s-b)(s-c)(s-d)}} \]

Where:

\( R \) — Circumradius of the cyclic quadrilateral
\( a, b, c, d \) — Sides of the cyclic quadrilateral
\( s \) — Semi-perimeter (\( s = \frac{P}{2} \))
\( P \) — Perimeter of the cyclic quadrilateral

Explanation: This formula calculates the radius of the circle that circumscribes a cyclic quadrilateral based on its side lengths and perimeter.

3. Importance of Circumradius Calculation

Details: Calculating the circumradius is important in geometry for understanding the properties of cyclic quadrilaterals, designing circular structures, and solving problems in engineering and architecture where circular patterns are involved.

4. Using the Calculator

Tips: Enter all four side lengths and the perimeter of the cyclic quadrilateral. All values must be positive numbers, and the perimeter must be greater than the sum of any three sides.

5. Frequently Asked Questions (FAQ)

Q1: What is a cyclic quadrilateral?
A: A cyclic quadrilateral is a four-sided polygon whose vertices all lie on a single circle.

Q2: Can any quadrilateral be cyclic?
A: No, only quadrilaterals that satisfy certain conditions can be cyclic. The opposite angles must sum to 180 degrees.

Q3: What are some examples of cyclic quadrilaterals?
A: Squares, rectangles, and isosceles trapezoids are all examples of cyclic quadrilaterals.

Q4: Why is the perimeter needed in the calculation?
A: The perimeter is used to calculate the semi-perimeter, which is a crucial component in the formula for determining the circumradius.

Q5: What units should I use for the inputs?
A: You can use any consistent unit of measurement (meters, centimeters, inches, etc.), but all inputs must use the same unit.