Sally works at "The Pentagon" building. Every morning she has to travel to different blocks to address the employees. She normally takes a circular path as her route.
Can you relate the concept of the "Circumscribe" to the path Sally takes every morning with respect to the Pentagon?
In this chapter, we will learn about the circumscribe in a circle, circumscribe definition, circumscribed and inscribed, circumscribed triangle along with the concept of circumscribe.
Check out the interactive examples to know more about the lesson and try your hand at solving a few interesting practice questions at the end of the page.
Lesson Plan
What is Meant by Circumscribe?
According to MerriamWebster, the meaning of circumscribe or circumscribing is to construct or be constructed around (a geometrical figure) so as to touch as many points as possible.
So any figure that circumscribes any other figure, means it is drawn outside another figure.
But the circumscribing figure must touch all the vertices of the inside figure.
For example, if a circle circumscribes a pentagon, it must touch all the 5 vertices of the pentagon.
Circle circumscribing pentagon
Explain Different Circumscribed Shapes
Any shape can circumscribe any other shape.
Circumscribed Shapes
Here are few circumscribed shapes with their illustrations:
Circumscribed Circle
The circumscribed circle is the circle drawn outside of any other shapes such as polygon, touching all the vertices of the polygon, and is termed as circumcircle.
Circle circumscribing triangle
Note: All 3 vertices have been touched by the circle.
Circumscribed Triangle
The circumscribed triangle is the triangle drawn outside of any other shapes. It must be touching all the vertices, if available.
Triangle circumscribing square
Circumscribed Pentagon
The circumscribed pentagon is the pentagon drawn outside of any other shapes. It must be touching all the vertices, if available.
Pentagon can be regular or irregular.
Pentagon circumscribing circle
Circumscribed Angle
The circumscribed angle is drawn when the arms of the angle touch the circle as tangents are drawn to the circle.
Angle circumscribing circle
Circumscribed Quadrilateral
The circumscribed quadrilateral is the quadrilateral drawn outside of any other shapes. It must be touching all the vertices, if available.
Quadrilateral circumscribing triangle
Circumscribed Hexagon
The circumscribed quadrilateral is the quadrilateral drawn outside of any other shapes. It must be touching all the vertices, if available.
Hexagon circumscribing circle
Circumscribed Rectangle
The circumscribed rectangle is the rectangle drawn outside of any other shapes. It must be touching all the vertices, if available.
Rectangle circumscribing triangle
What Is a Circumscribed Angle?
If we see an ice cream cone in 2D, we can see that the circular shape of the ice cream is perfectly fitting the cone's two sides.
The angle between the sides of the cone is known as the circumscribed angle.
The circumscribed angle is drawn when the arms of the angle touch the circle as tangents are drawn to the circle.
Example
Here is another example to relate to circumscribed angles in real life:
Football Trophies

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 that shape that is drawn inside another shape while a shape drawn outside another shape is referred to as a circumscribed shape.
Solved Examples
Example 1 
Nora wants to put a square curtain of each side 14 \( \text{in}\), completing enclosing her circular window.
Can you help her calculate the area of the inscribed circle?
Solution
Each side of the square curtain = 14 \( \text{in}\)
Radius of circle = \( \frac{14}{2} = 7\) \( \text{in}\)
Area of circle = \( \pi \times r^2\)
Thus, area of the circular window,
\( \begin{align*} \text {Area} &= \pi \times r^2 \\ &= \frac{22}{7} \times 7^2 = 154 \ \text {square in} \end{align*} \)
\( \therefore \text {Area} = 154 \ \text {square in}\) 
Example 2 
Lucy is solving the following question:
A quadrilateral \( \text {ABCD}\) 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 = \( 180^\circ \)
Therefore,
\( \begin{align*} \angle A + \angle C &= 180^\circ \\ \angle 78^\circ + \angle C &= 180^\circ \\ \angle C &= 180^\circ  \angle 78^\circ \\ \angle C &= 102^\circ \end{align*} \)
Similarly,
\( \begin{align*} \angle B + \angle D &= 180^\circ \\ \angle 105^\circ + \angle D &= 180^\circ \\ \angle D &= 180^\circ  \angle 105^\circ \\ \angle D &= 75^\circ \end{align*} \)
\( \therefore \begin{align*} \angle C &= 102^\circ \\ \angle D &= 75^\circ \end{align*} \) 
 How will you construct a triangle inside a circumscribed circle?
Interactive Questions
Here are a few activities for you to practice.
Select/type your answer and click the "Check Answer" button to see the result.
Let's Summarize
The minilesson targeted the fascinating concept of circumscribe. The math journey around the circumscribe formula starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.
About Cuemath
At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!
Through an interactive and engaging learningteachinglearning approach, the teachers explore all angles of a topic.
Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.
Frequently Asked Questions (FAQs)
1. What is the difference between circumscribed and inscribed?
An inscribed figure is that shape that is drawn inside another shape.
A shape drawn outside another shape is referred to as a circumscribed shape.
2. 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.
3. What is an inscribed circle?
An inscribed circle is the circle that is drawn inside any other figure.