Area of Cyclic Quadrilateral given Circumradius Calculator

Cyclic Quadrilateral Area Formula:

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

Side A of Cyclic Quadrilateral:

Side B of Cyclic Quadrilateral:

Side C of Cyclic Quadrilateral:

Side D of Cyclic Quadrilateral:

Circumradius of Cyclic Quadrilateral:

Area of Cyclic Quadrilateral:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is the Area of Cyclic Quadrilateral Formula?

The formula calculates the area of a cyclic quadrilateral (a quadrilateral inscribed in a circle) using its four sides and circumradius. This formula is derived from Brahmagupta's formula extended with the circumradius parameter.

2. How Does the Calculator Work?

The calculator uses the cyclic quadrilateral area formula:

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

Where:

\( S_a \) — Side A of the cyclic quadrilateral
\( S_b \) — Side B of the cyclic quadrilateral
\( S_c \) — Side C of the cyclic quadrilateral
\( S_d \) — Side D of the cyclic quadrilateral
\( r_c \) — Circumradius of the cyclic quadrilateral

Explanation: The formula combines the four sides and circumradius to compute the area through a sophisticated mathematical relationship that accounts for the cyclic nature of the quadrilateral.

3. Importance of Cyclic Quadrilateral Area Calculation

Details: Calculating the area of cyclic quadrilaterals is important in geometry, engineering design, architecture, and various mathematical applications where circularly inscribed quadrilaterals are encountered.

4. Using the Calculator

Tips: Enter all four side lengths and the circumradius in meters. All values must be positive numbers greater than zero for accurate calculation.

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 within the circle.

Q2: How does this formula differ from Brahmagupta's formula?
A: This formula incorporates the circumradius explicitly, while Brahmagupta's formula calculates area using only the sides and semiperimeter.

Q3: What are the units for the result?
A: The area is calculated in square meters (m²) if the input sides and circumradius are provided in meters.

Q4: Can this calculator handle decimal inputs?
A: Yes, the calculator accepts decimal values for more precise calculations.

Q5: What if the quadrilateral is not cyclic?
A: This formula only applies to cyclic quadrilaterals. For non-cyclic quadrilaterals, different area calculation methods must be used.