Circumradius of Cyclic Quadrilateral Given Area Calculator

Circumradius of Cyclic Quadrilateral Formula:

\[ R = \frac{\sqrt{((S_a \cdot S_b)+(S_c \cdot S_d)) \cdot ((S_a \cdot S_c)+(S_b \cdot S_d)) \cdot ((S_a \cdot S_d)+(S_c \cdot S_b))}}{4 \cdot A} \]

Side A of Cyclic Quadrilateral:

Side B of Cyclic Quadrilateral:

Side C of Cyclic Quadrilateral:

Side D of Cyclic Quadrilateral:

Area of Cyclic Quadrilateral:

m²

Circumradius of Cyclic Quadrilateral:

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1. What is the Circumradius of 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 circumradius formula:

\[ R = \frac{\sqrt{((S_a \cdot S_b)+(S_c \cdot S_d)) \cdot ((S_a \cdot S_c)+(S_b \cdot S_d)) \cdot ((S_a \cdot S_d)+(S_c \cdot S_b))}}{4 \cdot A} \]

Where:

\( R \) — Circumradius of the cyclic quadrilateral
\( S_a \) — Length of side A
\( S_b \) — Length of side B
\( S_c \) — Length of side C
\( S_d \) — Length of side D
\( A \) — Area of the cyclic quadrilateral

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

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 related to inscribed polygons in circles.

4. Using the Calculator

Tips: Enter all four side lengths and the area of the cyclic quadrilateral. All values must be positive numbers. The calculator will compute the circumradius of the circumscribed circle.

5. Frequently Asked Questions (FAQ)

Q1: What is a cyclic quadrilateral?
A: A cyclic quadrilateral is a four-sided polygon where all vertices lie on a single circle, making it inscribed in the circle.

Q2: Can any quadrilateral be cyclic?
A: No, only quadrilaterals where the sum of opposite angles equals 180 degrees can be cyclic (inscribed in a circle).

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

Q4: How is the area of a cyclic quadrilateral calculated?
A: The area can be calculated using Brahmagupta's formula when all four sides are known, or through other geometric methods depending on available information.

Q5: What practical applications does this calculation have?
A: This calculation is used in architecture, engineering design, computer graphics, and various geometric problem-solving scenarios involving circular patterns.