Circumscribed Circle Calculator

How the Circumscribed Circle Calculator Works

The Circumscribed Circle Calculator determines the properties of the circle that passes through all three vertices of a triangle. By entering the lengths of the triangle’s three sides, it calculates the radius, diameter, circumference, and area of the circumscribed circle, also known as the circumcircle.

This tool is essential for students of geometry, architects, engineers, and anyone involved in geometric construction or structural analysis. It's especially useful in projects where circular motion, rotational symmetry, or trigonometric relationships are important.

Formula

The radius R of the circumscribed circle is calculated using the formula:

R = (a × b × c) / (4 × Area)

Where:

• a, b, and c are the sides of the triangle

• Area is calculated using Heron’s formula:

s = (a + b + c) / 2
Area = √[s(s - a)(s - b)(s - c)]

Once the radius is found, the calculator provides:

• Diameter = 2 × R

• Circumference = 2 × π × R

• Area of circle = π × R²

Example Calculation

Input values:

• Side a = 15

• Side b = 5

• Side c = 12

Steps:

1. Semi-perimeter (s) = (15 + 5 + 12) / 2 = 16

2. Area (Heron) = √[16(16 - 15)(16 - 5)(16 - 12)] = √704 ≈ 26.53

3. Radius (R) = (15 × 5 × 12) / (4 × 26.53) ≈ 8.48 cm

4. Diameter = 2 × 8.48 = 16.96 cm

5. Circumference = 2 × π × 8.48 ≈ 53.28 cm

6. Area of circle = π × (8.48)² ≈ 225.91 cm²

Visual Interpretation

The circumscribed circle is the unique circle that touches all three vertices of a triangle. Its center is called the circumcenter, which lies:

• Inside the triangle (for acute triangles)

• On the hypotenuse (for right triangles)

• Outside the triangle (for obtuse triangles)

This makes the circumcircle fundamental in classical geometry and trigonometry.

Applications

This calculator is used in:

• Geometry: Triangle constructions, circle theorems

• Engineering: Structural and mechanical design

• Architecture: Design of circular layouts and elements

• Surveying: Triangulation and geodesic measurements

• CAD and 3D Modeling: Circle fitting and geometry constraints

Understanding the circumscribed circle helps ensure precision in rotational symmetry, design planning, and geometric analysis.

Why is this important?

The circumscribed circle allows you to:

• Determine relationships between side lengths and radius

• Apply trigonometric rules like the Law of Sines

• Analyze symmetry and rotation in mechanical systems

• Build circumscribed figures accurately in software and blueprints

It also supports mathematical proofs and optimization problems in design and architecture.

Can the triangle be any type?

Yes, as long as the triangle is valid (i.e., the sum of any two sides is greater than the third), the circumscribed circle always exists and is unique for every triangle.

• For equilateral triangles: the circumcenter is at the centroid

• For right triangles: the hypotenuse is the diameter of the circle

Can this be used with different units?

Absolutely. As long as all side lengths use the same unit (cm, meters, inches), the resulting radius, diameter, and area will be in corresponding units (e.g., cm, cm²). Just ensure consistency across all inputs.