Inscribed Quadrilaterals in Circles Calculator
Calculate area, circumradius, and angles of a cyclic quadrilateral.
Length of the first side of the quadrilateral.
Length of the second side.
Length of the third side.
Length of the fourth side.
Select the unit of measurement for the sides.
Results
Visual representation (not to scale)
What is an Inscribed Quadrilateral?
An inscribed quadrilateral, also known as a cyclic quadrilateral, is a four-sided polygon whose four vertices all lie on a single circle. This circle is called the circumcircle, and its radius is the circumradius. The key property of an inscribed quadrilaterals in circles calculator is that opposite angles sum to 180 degrees. This geometric constraint leads to several unique formulas for calculating its properties, such as area and diagonal lengths, based solely on its side lengths.
This calculator is useful for students, engineers, and designers who need to determine the geometric properties of such shapes without knowing the angles beforehand. The most famous formula associated with it is Brahmagupta’s formula for the area.
Inscribed Quadrilateral Formulas and Explanation
The calculations for a cyclic quadrilateral primarily rely on its side lengths (a, b, c, d). Our inscribed quadrilaterals in circles calculator uses the following key formulas:
Brahmagupta’s Formula for Area (K)
This formula calculates the area of a cyclic quadrilateral. It’s a generalization of Heron’s formula for triangles.
K = √[(s-a)(s-b)(s-c)(s-d)]
Where ‘s’ is the semi-perimeter: s = (a + b + c + d) / 2
Circumradius (R)
The radius of the circle that circumscribes the quadrilateral is calculated using this formula:
R = (1 / 4K) √[(ab + cd)(ac + bd)(ad + bc)]
Diagonals (p, q) and Ptolemy’s Theorem
The lengths of the two diagonals can be found as well. They are related by Ptolemy’s theorem example, which states ac + bd = pq. The individual diagonal lengths are:
p (diagonal AC) = √[((ac + bd)(ad + bc)) / (ab + cd)]
q (diagonal BD) = √[((ac + bd)(ab + cd)) / (ad + bc)]
| Variable | Meaning | Unit (auto-inferred) | Typical Range |
|---|---|---|---|
| a, b, c, d | Side lengths of the quadrilateral | cm, in, m, etc. | Positive numbers |
| s | Semi-perimeter | Same as sides | Positive number |
| K | Area | cm², in², m², etc. | Positive number |
| R | Circumradius | Same as sides | Positive number |
| A, B, C, D | Internal angles | Degrees | 0-180° |
Practical Examples
Example 1: A Simple Quadrilateral
- Inputs: a=7, b=8, c=9, d=10
- Units: cm
- Results:
- Area: 69.99 cm²
- Circumradius: 5.46 cm
- Angle A: 106.6°
- Angle C: 73.4°
Example 2: A Long, Thin Quadrilateral
- Inputs: a=3, b=15, c=5, d=14
- Units: inches
- Results:
- Area: 54.31 in²
- Circumradius: 8.76 in
- Angle A: 77.0°
- Angle C: 103.0°
How to Use This Inscribed Quadrilaterals in Circles Calculator
Using this tool is straightforward. Follow these simple steps:
- Enter Side Lengths: Input the lengths for the four consecutive sides of the quadrilateral (a, b, c, d).
- Check Validity: The calculator automatically checks if a cyclic quadrilateral can be formed with these side lengths. For a valid cyclic quadrilateral, the sum of any three sides must be greater than the fourth side.
- Select Units: Choose the appropriate unit of measurement from the dropdown menu. This ensures the results are labeled correctly.
- Interpret Results: The calculator instantly provides the primary result (Area) and secondary results like the circumradius, diagonal lengths, and all four internal angles. The results are updated in real-time as you type.
- Visualize: A simple SVG chart provides a visual representation of the quadrilateral’s shape inside its circumcircle.
For more detailed calculations, you might be interested in a cyclic quadrilateral area calculator.
Key Factors That Affect Inscribed Quadrilaterals
Several factors influence the properties calculated by the inscribed quadrilaterals in circles calculator:
- Side Length Ratios: The relative lengths of the sides dramatically alter the shape and area. A quadrilateral with nearly equal sides will be more ‘regular’ and often have a larger area for a given perimeter than a long, thin one.
- Perimeter: For a fixed set of side length ratios, a larger perimeter (scaling up all sides) will lead to a quadratically larger area and a linearly larger circumradius.
- Order of Sides: The calculated area will be the same regardless of the order you enter the sides, but the angles and diagonals will change. This calculator assumes sides a, b, c, and d are in sequence around the perimeter.
- Quadrilateral Inequality: A quadrilateral can only be formed if the sum of the lengths of any three sides is greater than the length of the fourth side. If this condition isn’t met, no such shape exists.
- Cyclic Condition: Not all quadrilaterals are cyclic. A quadrilateral can be inscribed in a circle only if its opposite angles are supplementary. This calculator assumes your side lengths can form such a quadrilateral.
- Unit Selection: While the numerical values of the calculations depend only on the side length numbers, selecting the correct units is crucial for interpreting the results in a real-world context. The area unit will be the square of the side unit. Our Brahmagupta’s formula calculator focuses specifically on area.
Frequently Asked Questions (FAQ)
- 1. What is a cyclic quadrilateral?
- A cyclic quadrilateral is another name for an inscribed quadrilateral—a four-sided figure whose vertices all lie on a circle.
- 2. Can any quadrilateral be inscribed in a circle?
- No. A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary (add up to 180 degrees).
- 3. What happens if I enter side lengths that can’t form a quadrilateral?
- The calculator will display an error message indicating that the side lengths do not satisfy the quadrilateral inequality theorem (the sum of any three sides must be greater than the fourth).
- 4. How is the area of an inscribed quadrilateral calculated?
- The calculator uses Brahmagupta’s formula, which finds the area using only the four side lengths. It is a powerful tool as it does not require angle or diagonal measurements.
- 5. What is the difference between an inscribed and a circumscribed circle?
- An inscribed quadrilateral has its vertices on a circle. A tangential quadrilateral has a circle inscribed within it, where the circle is tangent to all four sides. This calculator deals with the former.
- 6. Does the order of the side lengths matter?
- Yes. The sides should be entered in sequential order as you would travel around the perimeter (e.g., side ‘a’ is adjacent to ‘b’ and ‘d’). Changing the order can change the shape, its angles and diagonals, though the area remains the same.
- 7. How are the angles of the inscribed quadrilateral calculated?
- The angles are found using the law of cosines on the triangles formed by the diagonals. First, the diagonal lengths are calculated from the side lengths, then the angles can be determined. For a deeper dive, see our guide on angles of inscribed quadrilateral.
- 8. Why is Ptolemy’s Theorem important for cyclic quadrilaterals?
- Ptolemy’s Theorem provides a relationship between the side lengths and the diagonal lengths (ac + bd = pq). It’s a fundamental property unique to cyclic quadrilaterals and is used in the derivation of other formulas.