Area of Cyclic Quadrilateral Given Circumradius and Angles Calculator

Area of Cyclic Quadrilateral Formula:

\[ A = 2 \times (r_c)^2 \times \sin(\angle A) \times \sin(\angle B) \times \sin(\angle Diagonals) \]

Circumradius of Cyclic Quadrilateral:

Angle A of Cyclic Quadrilateral:

rad

Angle B of Cyclic Quadrilateral:

rad

Angle Between Diagonals of Cyclic Quadrilateral:

rad

Area of Cyclic Quadrilateral:

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1. What is the Area of Cyclic Quadrilateral Formula?

The area of a cyclic quadrilateral can be calculated using the formula that involves the circumradius and three angles of the quadrilateral. This formula provides an efficient way to determine the area when these specific parameters are known.

2. How Does the Calculator Work?

The calculator uses the area formula:

\[ A = 2 \times (r_c)^2 \times \sin(\angle A) \times \sin(\angle B) \times \sin(\angle Diagonals) \]

Where:

\( A \) — Area of Cyclic Quadrilateral (m²)
\( r_c \) — Circumradius of Cyclic Quadrilateral (m)
\( \angle A \) — Angle A of Cyclic Quadrilateral (rad)
\( \angle B \) — Angle B of Cyclic Quadrilateral (rad)
\( \angle Diagonals \) — Angle Between Diagonals of Cyclic Quadrilateral (rad)

Explanation: The formula calculates the area based on the circumradius and three specific angles of the cyclic quadrilateral, utilizing trigonometric functions for accurate computation.

3. Importance of Area Calculation

Details: Calculating the area of cyclic quadrilaterals is essential in various geometric applications, architectural design, and engineering projects where precise area measurements are required.

4. Using the Calculator

Tips: Enter the circumradius in meters, and all angles in radians. Ensure all values are positive and within valid ranges (angles between 0-2π radians).

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, known as the circumcircle.

Q2: Why use radians instead of degrees?
A: Radians are the standard unit for angular measurement in mathematical calculations involving trigonometric functions, providing more accurate results.

Q3: Can this formula be used for all cyclic quadrilaterals?
A: Yes, this formula applies to all cyclic quadrilaterals when the required parameters (circumradius and three angles) are known.

Q4: What if I have angles in degrees?
A: Convert degrees to radians by multiplying by π/180 before entering the values into the calculator.

Q5: Are there alternative methods to calculate the area?
A: Yes, other methods include using Brahmagupta's formula or dividing the quadrilateral into triangles, but this formula is specifically designed for cases where circumradius and angles are known.