Area of Quadrant

A quadrant is a quarter of a circle. It is formed by dividing a circle into four equal parts. The area of a quadrant is the total space enclosed by one of these parts.

A quadrant has a right angle (90 degrees) at its vertex, and its two radii are perpendicular to each other.

To find the area of a quadrant, we use the formula: \( \frac{1}{4} \times \pi \times r^2 \), where \( r \) is the radius of the circle. Let's see how the formula is derived.

Derivation of the formula:

1. The area of a full circle is given by \( \pi \times r^2 \).

2. A quadrant is one-fourth of a circle. Therefore, the area of a quadrant is \( \frac{1}{4} \times \pi \times r^2 \).

Therefore, the area of a quadrant = \( \frac{1}{4} \times \pi \times r^2 \).

We can find the area of a quadrant by using the formula with the radius of the circle. For example, if the radius \( r \) is provided, we use the formula.

For example, if the radius is 7 cm, what will be the area of the quadrant? Area = \( \frac{1}{4} \times \pi \times r^2 \) = \( \frac{1}{4} \times \pi \times 7^2 \) = \( \frac{1}{4} \times \pi \times 49 \) = \( \frac{49\pi}{4} \approx 38.48 \, \text{cm}^2 \) The area of the quadrant is approximately 38.48 cm².

We measure the area of a quadrant in square units. The measurement depends on the system used:

In the metric system, the area is measured in square meters (m²), square centimeters (cm²), and square millimeters (mm²).

In the imperial system, the area is measured in square inches (in²), square feet (ft²), and square yards (yd²).

Since a quadrant is a quarter of a circle, its area can be calculated using the radius. However, there are special cases:

1. If the diameter is given, halve it to find the radius before using the formula.

2. If the circumference is given, find the radius using \( \frac{\text{Circumference}}{2\pi} \).

3. Use \( \frac{1}{4} \times \pi \times r^2 \) for the area once the radius is known.

To ensure correct results while calculating the area of a quadrant, consider the following tips:

1. Always check if the given measurement is the diameter or radius.

2. Convert units if necessary to keep consistency (e.g., cm to m).

3. If given the circumference, use it to find the radius first before calculating the area.

• Area: The total space occupied by a shape. It tells us how much surface a shape covers.

• Quadrant: A quarter of a circle, formed by two radii at right angles.

• Radius: The distance from the center of a circle to any point on its perimeter.

• Circumference: The total distance around the circle.

• Pi (\( \pi \)): A mathematical constant approximately equal to 3.14159, representing the ratio of a circle's circumference to its diameter.