Angle Between Diagonals of Cyclic Quadrilateral Calculator

Angle Between Diagonals of Cyclic Quadrilateral Formula:

\[ \angle Diagonals = 2 \times \arctan\left(\sqrt{\frac{(s - Sb) \times (s - Sd)}{(s - Sa) \times (s - Sc)}}\right) \]

Semiperimeter of Cyclic Quadrilateral (s):

Side A of Cyclic Quadrilateral (Sa):

Side B of Cyclic Quadrilateral (Sb):

Side C of Cyclic Quadrilateral (Sc):

Side D of Cyclic Quadrilateral (Sd):

Angle Between Diagonals:

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1. What is the Angle Between Diagonals of Cyclic Quadrilateral?

The angle between diagonals of a cyclic quadrilateral is the angle formed at the intersection point of the two diagonals. This angle can be calculated using the semiperimeter and side lengths of the quadrilateral through a specific trigonometric formula.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \angle Diagonals = 2 \times \arctan\left(\sqrt{\frac{(s - Sb) \times (s - Sd)}{(s - Sa) \times (s - Sc)}}\right) \]

Where:

\( s \) — Semiperimeter of the cyclic quadrilateral
\( Sa \) — Side A length
\( Sb \) — Side B length
\( Sc \) — Side C length
\( Sd \) — Side D length

Explanation: The formula calculates the angle between diagonals using the semiperimeter and side lengths through trigonometric and inverse trigonometric functions.

3. Importance of Angle Calculation

Details: Calculating the angle between diagonals is important in geometry for understanding the properties of cyclic quadrilaterals, solving geometric problems, and applications in various fields including engineering and architecture.

4. Using the Calculator

Tips: Enter all side lengths and semiperimeter in meters. All values must be positive numbers, and the semiperimeter must be greater than each individual side length for valid 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.

Q2: Why is the semiperimeter important in this calculation?
A: The semiperimeter is used in the formula to relate the side lengths and calculate the angle between diagonals.

Q3: What units should I use for the inputs?
A: The calculator uses meters, but any consistent unit of length can be used as long as all inputs are in the same unit.

Q4: Are there any restrictions on the input values?
A: All side lengths must be positive, and the semiperimeter must be greater than each individual side length for the formula to be valid.

Q5: What does the result represent?
A: The result represents the angle in degrees between the two diagonals of the cyclic quadrilateral at their intersection point.