Diagonal 1 of Cyclic Quadrilateral using Ptolemy's Theorem Calculator

Ptolemy's Theorem Formula:

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

Side A of Cyclic Quadrilateral:

Side C of Cyclic Quadrilateral:

Side B of Cyclic Quadrilateral:

Side D of Cyclic Quadrilateral:

Diagonal 2 of Cyclic Quadrilateral:

Diagonal 1 of Cyclic Quadrilateral:

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1. What is Ptolemy's Theorem?

Ptolemy's Theorem states that for a cyclic quadrilateral, the sum of the products of the two pairs of opposite sides equals the product of the diagonals. This theorem provides a relationship between the sides and diagonals of a cyclic quadrilateral.

2. How Does the Calculator Work?

The calculator uses Ptolemy's Theorem formula:

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

Where:

\( Sa \) — Side A of the cyclic quadrilateral
\( Sc \) — Side C of the cyclic quadrilateral
\( Sb \) — Side B of the cyclic quadrilateral
\( Sd \) — Side D of the cyclic quadrilateral
\( d2 \) — Diagonal 2 of the cyclic quadrilateral
\( d1 \) — Diagonal 1 of the cyclic quadrilateral (calculated)

Explanation: The formula calculates the length of one diagonal when the lengths of all four sides and the other diagonal are known.

3. Importance of Diagonal Calculation

Details: Calculating diagonals in cyclic quadrilaterals is essential for geometric analysis, construction planning, and various engineering applications where precise measurements are required.

4. Using the Calculator

Tips: Enter all side lengths and the known diagonal 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 whose vertices all lie on a single circle.

Q2: Can this calculator be used for any quadrilateral?
A: No, this calculator specifically applies to cyclic quadrilaterals where Ptolemy's Theorem holds true.

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

Q4: 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.

Q5: Can I calculate Diagonal 2 using the same formula?
A: Yes, the formula can be rearranged to calculate either diagonal when the other diagonal and all four sides are known.