Inscribed Quadrilaterals in Circles Calculator
Calculate angles, side lengths, and properties of quadrilaterals inscribed in circles with precision
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Quadrilateral Visualization
Key Formulas for Inscribed Quadrilaterals
∠A + ∠C = 180°
∠B + ∠D = 180°
AC × BD = AB × CD + AD × BC
(Product of diagonals = Sum of products of opposite sides)
Area = √[(s-a)(s-b)(s-c)(s-d)]
where s = (a+b+c+d)/2
R = (1/4) × √[(ab+cd)(ac+bd)(ad+bc)] / Area
Properties of Inscribed Quadrilaterals
All four vertices lie on the circumference of a circle. This is the defining characteristic of inscribed quadrilaterals.
Opposite angles in an inscribed quadrilateral are supplementary (add up to 180°). This is a direct consequence of the inscribed angle theorem.
The product of the diagonals equals the sum of the products of opposite sides. This relationship is unique to cyclic quadrilaterals.
For a given perimeter, the inscribed quadrilateral has the maximum possible area among all quadrilaterals with the same side lengths.
The perpendicular bisectors of all four sides meet at the center of the circumscribed circle.
The diagonals of an inscribed quadrilateral divide it into four triangles, each inscribed in the same circle.
Common Types and Examples
Rectangle
A special case where all angles are 90°. The circumcenter coincides with the intersection of diagonals. Diagonal length equals the diameter of the circumscribed circle.
Isosceles Trapezoid
Has one pair of parallel sides and equal legs. The perpendicular from the center to each parallel side bisects that side. Base angles are equal.
Square
The most symmetric inscribed quadrilateral. All sides equal, all angles 90°. The circumradius equals (side length × √2) / 2.
General Cyclic Quadrilateral
Any quadrilateral that can be inscribed in a circle. Must satisfy the condition that opposite angles sum to 180°.
Step-by-Step Calculation Methods
1. Use the property that opposite angles sum to 180°
2. Calculate the fourth angle: D = 180° – B
3. Verify: A + C should equal 180°
1. Calculate semi-perimeter: s = (a+b+c+d)/2
2. Apply Brahmagupta’s formula for area
3. Use law of cosines to find angles
4. Calculate circumradius using the formula
1. Verify Ptolemy’s theorem: AC × BD = AB × CD + AD × BC
2. Use the relationship to find missing measurements
3. Apply trigonometric relations for angles