Diagonal 2 of Cyclic Quadrilateral using Ptolemy's Theorem Calculator

Formula Used:

\[ d2 = \frac{(Sa \times Sc) + (Sb \times Sd)}{d1} \]

Side A of Cyclic Quadrilateral:

Side C of Cyclic Quadrilateral:

Side B of Cyclic Quadrilateral:

Side D of Cyclic Quadrilateral:

Diagonal 1 of Cyclic Quadrilateral:

Diagonal 2 of Cyclic Quadrilateral:

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1. What is Diagonal 2 of Cyclic Quadrilateral?

Diagonal 2 of Cyclic Quadrilateral is a line segment joining opposite vertices (B and D) of the Cyclic Quadrilateral. In a cyclic quadrilateral (a quadrilateral inscribed in a circle), the diagonals have special properties that can be calculated using Ptolemy's Theorem.

2. How Does the Calculator Work?

The calculator uses Ptolemy's Theorem formula:

\[ d2 = \frac{(Sa \times Sc) + (Sb \times Sd)}{d1} \]

Where:

\( d2 \) — Diagonal 2 of Cyclic Quadrilateral (m)
\( Sa \) — Side A of Cyclic Quadrilateral (m)
\( Sc \) — Side C of Cyclic Quadrilateral (m)
\( Sb \) — Side B of Cyclic Quadrilateral (m)
\( Sd \) — Side D of Cyclic Quadrilateral (m)
\( d1 \) — Diagonal 1 of Cyclic Quadrilateral (m)

Explanation: Ptolemy's Theorem states that for a cyclic quadrilateral, the product of the diagonals equals the sum of the products of opposite sides. This formula rearranges the theorem to solve for the second diagonal.

3. Importance of Diagonal Calculation

Details: Calculating diagonals in cyclic quadrilaterals is crucial for geometric analysis, construction planning, and understanding the properties of inscribed polygons. It helps in determining various geometric relationships and solving complex geometric problems.

4. Using the Calculator

Tips: Enter all side lengths (A, B, C, D) and Diagonal 1 in meters. All values must be positive numbers greater than zero. The calculator will compute Diagonal 2 using Ptolemy's Theorem.

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 Ptolemy's Theorem important?
A: Ptolemy's Theorem provides a relationship between the sides and diagonals of a cyclic quadrilateral, making it a fundamental tool in Euclidean geometry.

Q3: Can this formula be used for any quadrilateral?
A: No, this formula specifically applies to cyclic quadrilaterals where all vertices lie on a circle.

Q4: What units should I use for the inputs?
A: The calculator works with any consistent unit of length, though meters are used as the default unit in the interface.

Q5: What if I get an error or unexpected result?
A: Ensure all input values are positive numbers and that the quadrilateral is indeed cyclic. The theorem only applies to cyclic quadrilaterals.