Calculate The Area of A Circle with 90 Degrees Removed

When you need to calculate the area of a circle with a 90-degree sector removed, you're essentially finding the area of a circle minus the area of a quarter-circle. This calculation is useful in various fields including geometry, engineering, and design where partial circle areas are needed.

What is the area of a circle with 90 degrees removed?

The area of a circle with 90 degrees removed refers to the area of a full circle minus the area of a quarter-circle (which is 90 degrees of a circle). This calculation is particularly useful when you need to determine the area of a circular segment or when working with circular designs that have a missing quarter.

Understanding this calculation helps in various practical scenarios such as determining the area of a pizza slice, calculating the area of a circular garden with a quarter removed, or designing circular patterns with missing sections.

Formula and calculation

The area of a circle with 90 degrees removed can be calculated using the following formula:

Formula

Area = πr² - (1/4)πr²

Where:

π (pi) is approximately 3.14159
r is the radius of the circle

This formula works by first calculating the area of the full circle (πr²) and then subtracting the area of a quarter-circle (which is (1/4)πr²). The result is the area of the circle with the 90-degree sector removed.

Note

This calculation assumes that the 90-degree sector is a quarter-circle. If the sector is not a perfect quarter-circle, additional calculations would be needed.

Worked example

Let's work through an example to see how this calculation works in practice.

Example Calculation

Suppose we have a circle with a radius of 5 units. We want to find the area of this circle with a 90-degree sector removed.

Calculate the area of the full circle: πr² = π × 5² = 25π ≈ 78.54 square units
Calculate the area of the quarter-circle: (1/4)πr² = (1/4) × 25π ≈ 19.63 square units
Subtract the quarter-circle area from the full circle area: 78.54 - 19.63 ≈ 58.91 square units

The area of the circle with 90 degrees removed is approximately 58.91 square units.

Practical Tip

When working with real-world measurements, it's often helpful to round the final answer to a reasonable number of decimal places based on the precision of your measurements.

Practical applications

The calculation of the area of a circle with 90 degrees removed has several practical applications across different fields:

Geometry and Design: Useful for calculating the area of circular designs with missing sections, such as circular logos or patterns.
Engineering: Helps in determining the area of circular components with missing sections, such as circular pipes or tanks.
Everyday Life: Useful for calculating the area of circular objects with missing sections, such as pizza slices or circular gardens.

Understanding this calculation can help you solve various real-world problems involving circular areas with missing sections.

FAQ

What is the difference between the area of a full circle and a circle with 90 degrees removed?

The area of a full circle is πr², while the area of a circle with 90 degrees removed is πr² - (1/4)πr². The difference is the area of the quarter-circle that has been removed.

Can I use this calculation for sectors that are not exactly 90 degrees?

No, this calculation specifically applies to sectors that are exactly 90 degrees. For other sector angles, you would need to use a different formula involving the sector angle.

How do I calculate the area of a circle with a different sector angle removed?

To calculate the area of a circle with a different sector angle removed, you would use the formula: Area = πr² - (θ/360)πr², where θ is the angle of the sector in degrees.