Circumscribed Circle Calculator

Circumradius (R): -

Center Coordinates: -

Area of Triangle: -

Understanding Circumscribed Circles

What is a Circumscribed Circle?

A circumscribed circle (circumcircle) is the unique circle that passes through all three vertices of a triangle. Key properties include:

  • Passes through all three vertices
  • Center (circumcenter) is equidistant from all vertices
  • Center is the intersection of perpendicular bisectors
  • Radius is called the circumradius
  • Can be inside, outside, or on the triangle (for right triangles)

Key Formulas

Circumradius Formula

R = abc/(4A)

where A is area and a,b,c are sides

Area Formula

A = āˆš(s(s-a)(s-b)(s-c))

Sine Law Relation

R = a/(2sin A) = b/(2sin B) = c/(2sin C)

Advanced Properties

Area Relationship

Area = abc/(4R)

Euler's Theorem

dĀ² = R(R - 2r)

where d is distance between centers

Nine-Point Circle

Radius is R/2

Ptolemy's Theorem

Relates to cyclic quadrilaterals

Special Cases

Equilateral Triangle

R = a/āˆš3 where a is side length

Right Triangle

R = c/2 where c is hypotenuse

Isosceles Triangle

Circumcenter lies on height to base

Obtuse Triangle

Circumcenter lies outside triangle

Real-World Applications

Surveying

Used in triangulation and mapping

Engineering

Applied in structural design and optimization

Computer Graphics

Essential in mesh generation and computational geometry

Astronomy

Used in celestial navigation and orbital mechanics