Circumscribe
A circumscribe or a circumcircle is a circle that passes through all the vertices of any polygon such as triangles, rectangles, regular polygons, and some other shapes but not all polygons. In other words, when a circle is drawn over another shape such as a triangle or rectangle while touching all corners or vertices of the shape inside is called the circumcircle or circumscribe. Let us learn more about this concept with the definition of circumcircle, the shapes circumscribed in another shape, the circumcircle formulas to solve a few examples.
1.  Circumcircle Definition 
2.  Circumcircle of Triangle 
3.  Steps to Construct a Circumcircle 
4.  Circumcircle Formula 
5.  Circumcircle or Circumscribed Circle Shapes 
6.  FAQs on Circumcircle 
Circumcircle Definition
The circumcircle of a polygon is defined as the circle that passes through all of its vertices or corners of a shape and all these polygons are known as cyclic polygons. However, not all polygons come under this criteria but only regular polygons, triangles, rectangles, and rightkites have circumcircles. The center of this circle is called the circumcenter i.e. the center or origin of the circle and its radius is called the circumradius i.e. the radius of the polygon's circumcircle. Look at the image below to understand it better.
Circumscribed Meaning
A circumcircle can also be referred to as circumscribing a polygon or circumscribed circle. Circumscribe or circumscribing is to construct or be constructed around a geometrical figure or polygon so as to touch as many points of the vertex as possible. Any figure is said to be circumscribed when one shape is drawn outside another shape touching the corners. For example, if a circle circumscribes a pentagon, it must touch all the 5 vertices of the pentagon.
Circumcircle of Triangle
The circumcircle of a triangle is defined as a circle passing through all the three vertices or corners of the triangle. The center is the point where all the perpendicular bisectors of the triangle's sides meet formed while constructing the circumcircle is called the circumcenter. The center can be both inside and outside of the triangle. The line segment from the circumcenter to any point of the circumcircle is called the circumradius. For a right triangle, the diameter of the circumcircle is the hypotenuse of the triangle and the center or origin is the midpoint of the hypotenuse.
Steps to Construct a Circumcircle
For constructing a circumcircle of a triangle, we need to find construct perpendicular bisectors on either side of the triangle that intersects at a point called the circumcenter of the circumcircle. The three simple steps of construction are:
 Step 1: Construct a triangle with the given angle measurements.
 Step 2: Construct a perpendicular bisector on either side of the triangle keeping the base of the triangle as the line segment to construct it.
 Step 3: Intersect both the lines at a point making it the origin or center of the circumcircle.
 Step 4: Using the center, draw a circle around the triangle touching all the three vertices or corners of the triangle.
Let us look at an example. Construct a circumcircle of a triangle ABC with AB = 6cm, ∠A = 60° and ∠B = 60°.
 Step 1: Construct triangle ABC with the base line segment as AB = 6cm, ∠A = 60° and ∠B = 60°
 Step 2: Construct the perpendicular bisects of the triangle ABC.
 Step 3: Intersect both the perpendicular lines creating the center as O.
 Step 4: Keeping O as the center, draw a circumcircle around the triangle ABC.
Circumcircle Formula
The circumcircle formula i.e. the area and the perimeter of the shape is similar to the formula of a circle. If a circle has radius r, then the formulas for the area and perimeter of that circle, are as follows:
 Area of a circle = πr^{2}
 Perimeter of a circle = 2πr
Circumcircle or Circumscribed Circle Shapes
Any shape can circumscribe any other shape. Here are a few circumscribed shapes or circumcircles with their illustrations:
Circumscribed Shape  Image 

Circumscribed pentagon is the pentagon drawn outside of any other shape. It must be touching all the vertices, if available. Pentagon can be regular or irregular. 

Circumscribed angle is drawn when the arms of the angle touch the circle as tangents are drawn to the circle.  
Circumscribed quadrilateral is the quadrilateral drawn outside of any other shapes. It must be touching all the vertices, if available.  
Circumscribed rectangle is a rectangle drawn outside of any other shapes. It must be touching all the vertices, if available. 
Important Notes
 Circumcircle is also referred to as circumscribed circle.
 Not all quadrilaterals are circumscribed quadrilaterals.
 If any shape is circumscribing any other shape, it must be touching all the vertices, if available.
 The inscribed and circumscribed figure are complementary terms. An inscribed figure is a shape that is drawn inside another shape while a shape drawn outside another shape is referred to as a circumscribed shape.
Circumcircle Examples

Example 1: Nora wants to put a square curtain on each side 14 in, completing enclosing her circular window. Can you help her calculate the area of the inscribed circle?
Solution:
Each side of the square curtain = 14 in
Radius of circle = 14/2 = 7 in
Area of circle = πr^{2}
Thus, area of the circular window,
Area = πr^{2}
Area = 22/7 × (7)^{2}
Area = 154
Therefore, area of the inscribed circle is 154 square in.

Example 2: Lucy is solving the question  A quadrilateral ABCD is inscribed in a circle as shown in the figure. Can you help her determine the unknown angles?
Solution: We know,
For a quadrilateral inscribed in a circle, the sum of opposite sides = 180°
Therefore,
∠A + ∠C = 180°
∠78°+∠C = 180°
∠C = 180°−∠78°
∠C =102°
Similarly,
∠B +∠D = 180°
∠105°+∠D = 180°
∠D = 180°−∠105°
∠D = 75°
Therefore, ∠C =102° and ∠D = 75°.
FAQs on Circumcircle
What is a Circumcircle?
A circumcircle is a circle that passes all the vertices of a regular polygon.
What is the Difference Between Circumscribed and Inscribed?
An inscribed figure is a shape that is drawn inside another shape. A shape drawn outside another shape is referred to as a circumscribed shape.
How Do You Find the Radius of a Circumscribed Circle?
The radius of a circumscribed circle can be calculated by comparing the dimensions of both the inscribed figure and the circumscribed circle.
What is an Inscribed Circle?
An inscribed circle is the circle that is drawn inside any other figure.
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