Calculating the Area of Each Sector

Oct 13, 2025 by wbrum.org 2 views

Assertion: A Circle Circumscribes a Square of Side a, Then the Area of the Region Outside the Square and Enclosed by the Circle is [(pi/2)-1] a^2 Square Units

Reason: A Circle is Inscribed in a Square of Side 2a Units. If the Circle Touches All

Discussion Category: CBSE Board X

Introduction

In geometry, circles and squares are two fundamental shapes that have been studied extensively. A circle is a set of points that are equidistant from a central point called the center, while a square is a quadrilateral with four equal sides and four right angles. In this article, we will explore the relationship between a circle and a square, specifically when a circle circumscribes a square of side 'a' units. We will also discuss the area of the region outside the square and enclosed by the circle.

Understanding the Problem

To begin with, let's understand the problem statement. A circle circumscribes a square of side 'a' units, which means that the circle passes through all the vertices of the square. The problem asks us to find the area of the region outside the square and enclosed by the circle.

Visualizing the Problem

To visualize the problem, let's draw a diagram. We have a square of side 'a' units, and a circle that circumscribes the square. The circle passes through all the vertices of the square, and the region outside the square and enclosed by the circle is the area we need to find.

Finding the Area

To find the area of the region outside the square and enclosed by the circle, we need to use the concept of sectors and triangles. Let's consider the square as a whole, and the circle as a set of sectors. Each sector is a portion of the circle that is bounded by two radii and an arc.

Calculating the Area of Each Sector

The area of each sector can be calculated using the formula:

Area of sector = (θ/360) × πr^2

where θ is the central angle of the sector, and r is the radius of the circle.

Calculating the Area of the Triangle

The area of the triangle can be calculated using the formula:

Area of triangle = (base × height) / 2

In this case, the base of the triangle is 'a' units, and the height of the triangle is 'a' units.

Finding the Total Area

To find the total area of the region outside the square and enclosed by the circle, we need to add the areas of all the sectors and triangles.

Using the Formula

The formula for the area of the region outside the square and enclosed by the circle is:

Area = [(π/2) - 1]a^2

This formula can be derived by using the concept of sectors and triangles, and by adding the areas of all the sectors and triangles.

Conclusion

In conclusion, when a circle circumscribes a square of side 'a' units, the area of the region outside the square and enclosed by the circle is [(π/2) - 1]a^2 square units. This formula can be derived by using the concept of sectors and triangles, and by adding the areas of all the sectors and triangles.

Key Takeaways

A circle circumscribes a square of side 'a' units.
The area of the region outside the square and enclosed by the circle is [(π/2) - 1]a^2 square units.
The formula can be derived by using the concept of sectors and triangles, and by adding the areas of all the sectors and triangles.

Practice Questions

A circle circumscribes a square of side 'b' units. Find the area of the region outside the square and enclosed by the circle.
A circle is inscribed in a square of side 'c' units. Find the area of the region outside the square and enclosed by the circle.

Answer Key

The area of the region outside the square and enclosed by the circle is [(π/2) - 1]b^2 square units.
The area of the region outside the square and enclosed by the circle is [(π/2) - 1]c^2 square units.

References

[1] Geometry by Michael S. Klamkin
[2] Mathematics for CBSE Class X by R. S. Aggarwal

Note

The above article is for educational purposes only and is not intended to be used as a substitute for professional advice or guidance.
Q&A: A Circle Circumscribes a Square of Side a, Then the Area of the Region Outside the Square and Enclosed by the Circle is [(pi/2)-1] a^2 Square Units

Frequently Asked Questions

Q1: What is the relationship between a circle and a square when a circle circumscribes a square of side 'a' units?

A1: When a circle circumscribes a square of side 'a' units, the circle passes through all the vertices of the square.

Q2: How do we find the area of the region outside the square and enclosed by the circle?

A2: To find the area of the region outside the square and enclosed by the circle, we need to use the concept of sectors and triangles. We can calculate the area of each sector using the formula: Area of sector = (θ/360) × πr^2, and the area of the triangle using the formula: Area of triangle = (base × height) / 2.

Q3: What is the formula for the area of the region outside the square and enclosed by the circle?

A3: The formula for the area of the region outside the square and enclosed by the circle is: Area = [(π/2) - 1]a^2.

Q4: Can we derive the formula for the area of the region outside the square and enclosed by the circle using the concept of sectors and triangles?

A4: Yes, we can derive the formula for the area of the region outside the square and enclosed by the circle using the concept of sectors and triangles. By adding the areas of all the sectors and triangles, we can get the total area of the region outside the square and enclosed by the circle.

Q5: What is the significance of the formula [(π/2) - 1]a^2 in the context of the problem?

A5: The formula [(π/2) - 1]a^2 represents the area of the region outside the square and enclosed by the circle. It is a mathematical expression that gives us the exact value of the area.

Q6: Can we apply the formula [(π/2) - 1]a^2 to find the area of the region outside the square and enclosed by the circle for any value of 'a'?

A6: Yes, we can apply the formula [(π/2) - 1]a^2 to find the area of the region outside the square and enclosed by the circle for any value of 'a'. The formula is a general expression that can be used for any value of 'a'.

Q7: What are the key takeaways from the problem and its solution?

A7: The key takeaways from the problem and its solution are:

A circle circumscribes a square of side 'a' units.
The area of the region outside the square and enclosed by the circle is [(π/2) - 1]a^2 square units.
The formula can be derived by using the concept of sectors and triangles, and by adding the areas of all the sectors and triangles.

Q8: Can we use the formula [(π/2) - 1]a^2 to solve other problems related to circles and squares?

A8: Yes, we can use the formula [(π/2) - 1]a^2 to solve other problems related to circles and squares. The formula is a general expression that can be used to find the area of the region outside the square and enclosed by the circle for any value of 'a'.

Q9: What are the limitations of the formula [(π/2) - 1]a^2?

A9: The formula [(π/2) - 1]a^2 is a mathematical expression that gives us the exact value of the area of the region outside the square and enclosed by the circle. However, it is limited to the specific case where the circle circumscribes a square of side 'a' units.

Q10: Can we use the formula [(π/2) - 1]a^2 to find the area of the region outside the square and enclosed by the circle for a square of side 'b' units?

A10: Yes, we can use the formula [(π/2) - 1]a^2 to find the area of the region outside the square and enclosed by the circle for a square of side 'b' units. We just need to substitute 'b' for 'a' in the formula.

Conclusion

In conclusion, the formula [(π/2) - 1]a^2 is a mathematical expression that gives us the exact value of the area of the region outside the square and enclosed by the circle. It can be derived by using the concept of sectors and triangles, and by adding the areas of all the sectors and triangles. The formula is a general expression that can be used to find the area of the region outside the square and enclosed by the circle for any value of 'a'.