Cyclic Quadrilateral (Brahmagupta's) Formula Calculator

\[ A = \sqrt{(s-a)(s-b)(s-c)(s-d)} \quad \]

Side Length \(a\):

Side Length \(b\):

Side Length \(c\):

Side Length \(d\):

Area:

Semi-perimeter:

1. What is the Cyclic Quadrilateral (Brahmagupta's) Formula Calculator?

Definition: This calculator computes the area (\(A\)) and semi-perimeter (\(s\)) of a cyclic quadrilateral using Brahmagupta’s formula. A cyclic quadrilateral is a four-sided shape where all vertices lie on a single circle.

Purpose: It is used in geometry to determine the area of cyclic quadrilaterals, such as in architecture, land surveying, and mathematical studies.

2. How Does the Calculator Work?

The calculator uses the following formulas:

Formulas: \[ A = \sqrt{(s-a)(s-b)(s-c)(s-d)} \quad \text{where} \quad s = \frac{a + b + c + d}{2} \] where:

\(A\): Area (m², km², cm², mm²)
\(s\): Semi-perimeter (m, km, cm, mm)
\(a, b, c, d\): Side lengths (m, km, cm, mm)

Unit Conversions:

Side Lengths and Semi-perimeter:
- 1 m = 1 m
- 1 km = 1000 m
- 1 cm = 0.01 m
- 1 mm = 0.001 m
Area:
- 1 m² = 1 m²
- 1 km² = 1,000,000 m²
- 1 cm² = 0.0001 m²
- 1 mm² = 0.000001 m²

Steps:

Enter the side lengths \(a, b, c, d\) in m, km, cm, or mm (default values are 3, 4, 5, 6 m, step size 0.00001).
Convert all side lengths to meters.
Validate that all side lengths are positive and form a cyclic quadrilateral.
Calculate the semi-perimeter: \(s = \frac{a + b + c + d}{2}\).
Calculate the area: \(A = \sqrt{(s-a)(s-b)(s-c)(s-d)}\).
Convert the area and semi-perimeter to the selected output units.
Display the area and semi-perimeter, using scientific notation if the absolute value is less than 0.001, otherwise rounded to 2 decimal places.

3. Importance of Cyclic Quadrilateral Calculation

Calculating the area of a cyclic quadrilateral is crucial for:

Geometry and Trigonometry: Understanding properties of cyclic shapes in mathematical proofs and applications.
Land Surveying: Measuring the area of irregularly shaped plots that can be inscribed in a circle.
Architecture and Design: Designing structures with cyclic quadrilateral elements, such as arched windows or pavilions.

4. Using the Calculator

Examples:

Example 1: Calculate the area and semi-perimeter of a cyclic quadrilateral with side lengths \(a = 3 \, \text{m}\), \(b = 4 \, \text{m}\), \(c = 5 \, \text{m}\), \(d = 6 \, \text{m}\), with area in m² and semi-perimeter in m:
- Enter \(a = 3 \, \text{m}\), \(b = 4 \, \text{m}\), \(c = 5 \, \text{m}\), \(d = 6 \, \text{m}\).
- Semi-perimeter: \(s = \frac{3 + 4 + 5 + 6}{2} = 9 \, \text{m}\).
- Area: \(A = \sqrt{(9-3)(9-4)(9-5)(9-6)} = \sqrt{6 \times 5 \times 4 \times 3} = \sqrt{360} \approx 18.97 \, \text{m}^2\).
- Result: \( \text{Area} = 18.97 \, \text{m}^2 \), \( \text{Semi-perimeter} = 9.00 \, \text{m} \).
Example 2: Calculate the area and semi-perimeter of a cyclic quadrilateral with side lengths \(a = 1000 \, \text{mm}\), \(b = 1000 \, \text{mm}\), \(c = 1000 \, \text{mm}\), \(d = 1000 \, \text{mm}\), with area in cm² and semi-perimeter in cm:
- Enter \(a = 1000 \, \text{mm}\), \(b = 1000 \, \text{mm}\), \(c = 1000 \, \text{mm}\), \(d = 1000 \, \text{mm}\).
- Convert to meters: \(1000 \, \text{mm} = 1 \, \text{m}\).
- Semi-perimeter: \(s = \frac{1 + 1 + 1 + 1}{2} = 2 \, \text{m} = 200 \, \text{cm}\).
- Area: \(A = \sqrt{(2-1)(2-1)(2-1)(2-1)} = \sqrt{1 \times 1 \times 1 \times 1} = 1 \, \text{m}^2 = 10000 \, \text{cm}^2\).
- Result: \( \text{Area} = 10000.00 \, \text{cm}^2 \), \( \text{Semi-perimeter} = 200.00 \, \text{cm} \).

5. Frequently Asked Questions (FAQ)

Q: What is a cyclic quadrilateral?
A: A cyclic quadrilateral is a four-sided shape where all four vertices lie on a single circle.

Q: What does the semi-perimeter represent?
A: The semi-perimeter is half the perimeter of the quadrilateral, used in Brahmagupta’s formula to compute the area.

Q: Why might the calculator show an error?
A: An error occurs if the side lengths are not positive or if they cannot form a cyclic quadrilateral (e.g., the term under the square root becomes negative).