Circumscribed Circle Calculator
Calculate circumradius, circumcenter, and properties of circles circumscribed about triangles
Circumscribed Circle Calculator
Circumscribed Circle Diagram
Example Calculation
Garden Design Example
Problem: A triangular garden has sides of 6m, 8m, and 10m.
Question: What is the radius of the smallest circle that can contain this garden?
This circle is the circumscribed circle of the triangular garden.
Solution
Given: a = 6m, b = 8m, c = 10m
Check triangle inequality: 6 + 8 = 14 > 10 ✓, 6 + 10 = 16 > 8 ✓, 8 + 10 = 18 > 6 ✓
Note: This is a right triangle since 6² + 8² = 36 + 64 = 100 = 10²
For right triangles: R = hypotenuse / 2 = 10 / 2 = 5m
Verification with general formula:
• Semiperimeter: s = (6 + 8 + 10) / 2 = 12m
• Area: A = √[12(12-6)(12-8)(12-10)] = √[12×6×4×2] = √576 = 24m²
• Circumradius: R = (6×8×10) / (4×24) = 480 / 96 = 5m ✓
Key Concepts
Circumscribed Circle
Circle passing through all triangle vertices
Also called circumcircle
Circumcenter
Center of circumscribed circle
Intersection of perpendicular bisectors
Circumradius
Radius of circumscribed circle
R = (abc) / (4A)
Formula Reference
General Formula
R = (a × b × c) / (4 × A)
Area (Heron's Formula)
A = √[s(s-a)(s-b)(s-c)]
where s = (a+b+c)/2
Right Triangle
R = hypotenuse / 2
Equilateral Triangle
R = a / √3
Circle Properties
Diameter = 2R
Circumference = 2πR
Area = πR²
Understanding Circumscribed Circles
What is a Circumscribed Circle?
A circumscribed circle (or circumcircle) is a circle that passes through all vertices of a triangle. Every triangle has exactly one circumscribed circle. The center of this circle is called the circumcenter, and its radius is called the circumradius.
Properties of Circumscribed Circles
- •Every triangle has exactly one circumscribed circle
- •The circumcenter is equidistant from all three vertices
- •For right triangles, the circumcenter is at the midpoint of the hypotenuse
- •For acute triangles, the circumcenter is inside the triangle
- •For obtuse triangles, the circumcenter is outside the triangle
Real-World Applications
- •Architecture and structural design
- •Landscape and garden design
- •Engineering and construction planning
- •Computer graphics and geometric modeling
- •Navigation and GPS triangulation
Construction Tip: To construct a circumscribed circle, draw the perpendicular bisectors of any two sides of the triangle. Their intersection point is the circumcenter. Draw a circle from this center through any vertex.
Special Triangle Cases
Right Triangle
Circumcenter at hypotenuse midpoint
Formula: R = c/2
where c is the hypotenuse
Circle diameter = hypotenuse length
Equilateral Triangle
Circumcenter = centroid = orthocenter
Formula: R = a/√3
where a is the side length
Circumcenter inside triangle
Obtuse Triangle
Circumcenter outside triangle
Largest circumradius for given perimeter
Use general formula R = abc/(4A)
Longest side < diameter