⭕ Circumscribed Circle Calculator
Calculate the circle that circumscribes a triangle, square, or rectangle
How to Use This Calculator
Select Shape
Choose whether you want to circumscribe a triangle, square, or rectangle.
Enter Dimensions
Input the side lengths or dimensions of the shape. For a triangle, enter all three sides. For a square, enter one side. For a rectangle, enter length and width.
Calculate
Click "Calculate" to find the radius, diameter, circumference, and area of the circumscribed circle.
Review Results
See the circumradius (radius of circumscribed circle) and all circle properties displayed clearly.
Formula
Triangle
R = (a × b × c) / (4 × Area)
Where R is the circumradius, a, b, c are triangle sides, and Area is calculated using Heron's formula.
Heron's Formula: Area = √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2
Square
R = (Side × √2) / 2 = Side / √2
Where R is the circumradius and Side is the length of one side of the square. The circumradius equals half the diagonal length.
Rectangle
R = Diagonal / 2 = √(Length² + Width²) / 2
Where R is the circumradius, and Diagonal is calculated using the Pythagorean theorem.
Circle Properties:
- Diameter = 2 × Radius
- Circumference = 2 × π × Radius
- Area = π × Radius²
About Circumscribed Circle Calculator
The Circumscribed Circle Calculator finds the circle that passes through all vertices of a triangle, square, or rectangle. This circle is called the circumcircle, and its center is the circumcenter. The radius of this circle is called the circumradius.
When to Use This Calculator
- Geometry: Find the circumcircle of polygons
- Construction: Design circular structures around triangular or rectangular bases
- Education: Understand circumscribed circles and circumradius
- Engineering: Calculate dimensions for circular components
- Architecture: Design circular elements around geometric shapes
Why Use Our Calculator?
- ✅ Multiple Shapes: Works with triangles, squares, and rectangles
- ✅ Complete Results: Shows radius, diameter, circumference, and area
- ✅ Accurate Calculations: Uses precise mathematical formulas
- ✅ Educational: Helps understand circumscribed circles
- ✅ Free: No registration required
Key Concepts
- Circumscribed Circle: A circle that passes through all vertices of a polygon
- Circumradius: The radius of the circumscribed circle
- Circumcenter: The center of the circumscribed circle
- For Triangles: All triangles have a unique circumcircle
- For Rectangles: All rectangles (including squares) have a circumcircle with radius equal to half the diagonal
Example
For a triangle with sides 5, 6, and 7:
- Using Heron's formula: s = 9, Area ≈ 14.7
- Circumradius R = (5 × 6 × 7) / (4 × 14.7) ≈ 3.57
- Diameter = 2 × 3.57 = 7.14
- Circumference ≈ 22.44
Frequently Asked Questions
What is a circumscribed circle?
A circumscribed circle (circumcircle) is a circle that passes through all vertices of a polygon. For a triangle, square, or rectangle, there is exactly one such circle.
How do you find the circumradius of a triangle?
For a triangle, the circumradius R = (a × b × c) / (4 × Area), where a, b, c are the side lengths and Area is calculated using Heron's formula.
What's the circumradius of a square?
For a square with side length s, the circumradius is s / √2, which equals half the diagonal length.
Can all polygons have a circumscribed circle?
No, only cyclic polygons have a circumscribed circle. All triangles, all rectangles (including squares), and regular polygons are cyclic.
What's the difference between circumscribed and inscribed circle?
A circumscribed circle goes around a polygon (passes through vertices). An inscribed circle (incircle) goes inside a polygon (tangent to sides).
How is circumradius different from inradius?
Circumradius is the radius of the circle circumscribed around a polygon. Inradius is the radius of the circle inscribed inside a polygon. For most shapes, circumradius ≥ inradius.