Corner Quadrants of a Square and Central Circle | Class 10

Quick answer: Four equal quadrants (quarter circles) of radius $r$ together make one full circle of area $\pi r^2$. So if a quadrant of radius $1\text{ cm}$ is cut from each corner of a $4\text{ cm}$ square and a circle of diameter $2\text{ cm}$ is cut from the centre, the remaining area is $16 - \frac{22}{7} - \frac{22}{7} = \frac{68}{7} \approx 9.71\text{ cm}^2$. For four touching circles of radius $7\text{ cm}$, the area enclosed between them is $196 - 154 = 42\text{ cm}^2$.

Many area questions involve a shape with parts cut out of it. The method is always the same: find the area of the whole shape, find the area of the parts removed, and subtract.

When quadrants are cut from the corners of a square, there is a useful shortcut: four quadrants of the same radius together make one full circle.

In CBSE Class 10 Mathematics, Chapter 11 (Areas Related to Circles), problems with corner quadrants and circles inside squares are common board exam questions.

What You Will Learn

• How to find the area of a shape with parts cut out
• Why four quadrants make one full circle
• Problem 1: the NCERT problem of a square with four corner quadrants and a central circle
• Problem 2: the area enclosed between four touching circles
• Common mistakes with radius and diameter

1. Four Quadrants Make One Circle

A quadrant is a sector of a circle with a central angle of $90^\circ$, i.e. a quarter of a circle.

$\text{Area of one quadrant} = \frac{90^\circ}{360^\circ} \times \pi r^2 = \frac{1}{4}\pi r^2$

$\text{Area of four quadrants} = 4 \times \frac{1}{4}\pi r^2 = \pi r^2$

Key point: Four quadrants of the same radius have the same total area as one full circle of that radius.

Each corner of a square is a right angle ($90^\circ$), so a circle centred at a corner gives a quadrant inside the square.

2. Problem 1: Corner Quadrants and a Central Circle (NCERT)

Question

From each corner of a square of side $4\text{ cm}$, a quadrant of a circle of radius $1\text{ cm}$ is cut, and a circle of diameter $2\text{ cm}$ is also cut from the centre. Find the area of the remaining portion of the square. (Use $\pi = \frac{22}{7}$)

  +--.---------------.--+
  | /                 \ |
  |'                   '|
  |        .---.        |
  |       (  O  )       |
  |        '---'        |
  |.                   .|
  | \                 / |
  +--'---------------'--+
     side of square = 4 cm

Solution

Step 1: Area of the square.

$\text{Area of square} = 4^2 = 16\text{ cm}^2$

Step 2: Area of the four corner quadrants. Each has radius $1\text{ cm}$, and together they make one circle:

$\text{Area of 4 quadrants} = \pi r^2 = \frac{22}{7} \times 1^2 = \frac{22}{7}\text{ cm}^2$

Step 3: Area of the central circle. Its diameter is $2\text{ cm}$, so its radius is $1\text{ cm}$:

$\text{Area of circle} = \frac{22}{7} \times 1^2 = \frac{22}{7}\text{ cm}^2$

Step 4: Total area removed.

$\frac{22}{7} + \frac{22}{7} = \frac{44}{7}\text{ cm}^2$

Step 5: Remaining area.

$16 - \frac{44}{7} = \frac{112 - 44}{7} = \frac{68}{7}\text{ cm}^2 \approx 9.71\text{ cm}^2$

Answer: The area of the remaining portion is $\frac{68}{7}\text{ cm}^2$, or about $9.71\text{ cm}^2$.

3. Problem 2: Area Between Four Touching Circles

Question

Four equal circles, each of radius $7\text{ cm}$, are drawn so that each circle touches two of the others externally. Find the area of the region enclosed between them. (Use $\pi = \frac{22}{7}$)

Solution

Step 1: Join the centres. Joining the centres of the four circles forms a square. Its side is the sum of two radii:

$\text{Side} = 7 + 7 = 14\text{ cm}$

Step 2: Area of the square.

$14^2 = 196\text{ cm}^2$

Step 3: Area of the four quadrants inside the square. At each corner of the square, a quarter of one circle lies inside the square. Together, these four quadrants make one full circle of radius $7\text{ cm}$:

$\pi r^2 = \frac{22}{7} \times 7^2 = 154\text{ cm}^2$

Step 4: Enclosed area.

$196 - 154 = 42\text{ cm}^2$

Answer: The area enclosed between the four circles is $42\text{ cm}^2$.

4. Summary Table

5. Exam Tips and Common Mistakes

Exam Tip: Use the shortcut four quadrants = one circle to save time, but write it as a step so the examiner can follow your working.

Exam Tip: In Problem 1, the central circle has the same radius ($1\text{ cm}$) as the quadrants, so the total area removed is $2\pi r^2$.

Common Mistake: Using the diameter as the radius. A circle of diameter $2\text{ cm}$ has radius $1\text{ cm}$. Using $r = 2\text{ cm}$ makes its area four times too large.

Common Mistake: In the four-circle problem, using the radius as the side of the square. The side is two radii, $14\text{ cm}$.

Frequently Asked Questions

What is the area of a quadrant of a circle?

A quadrant is a quarter of a circle, so its area is ¼πr².

Why do four quadrants make a full circle?

Each quadrant has a central angle of 90°, and four of them add up to 360°, which is a full circle.

What is the remaining area when four corner quadrants and a central circle are cut from a 4 cm square?

16 − 22/7 − 22/7 = 68/7 cm², which is about 9.71 cm².

What is the area enclosed between four touching circles of radius 7 cm?

The square joining their centres has area 196 cm², and the four quadrants inside it have area 154 cm², so the enclosed area is 42 cm².