Circular sector Lesson
Degree, radian, and gradian are units of measurement used to describe angles in mathematics and geometry.
- Degree: A degree (°) is a unit of measurement for angles. One complete revolution around a circle is equal to 360 degrees.
- Radian: A radian is a unit of measurement for angles, defined as the ratio of the arc length of a circle to its radius. One radian is equal to approximately 57.3 degrees.
- Gradian: A gradian (grad) is a unit of measurement for angles, defined as 1/400 of a complete revolution. One gradian is equal to 9/10 of a degree. The gradian is not as commonly used as degrees and radians in modern mathematics and is mostly used in older European systems of measurement.
In general, degrees and radians are the most commonly used units of measurement for angles, with degrees being more commonly used in everyday applications and radians being more commonly used in mathematical and scientific calculations.
Circular Sector
A circular sector is a portion of a circle enclosed by two radii and an arc. It is a type of geometric shape that is created by taking a slice of a circle. The circular sector is often used to represent a fraction of the total area of a circle, and its size can be determined by specifying the central angle of the sector.
A circular sector is characterized by two parameters:
area
The area of a circular sector can be calculated using the following formula:
A = (θ/360)πr^2
where θ is the central angle of the sector in degrees, π is the mathematical constant pi (approximately equal to 3.14), and r is the radius of the circle.
Alternatively, the formula can be written in terms of the arc length s of the sector, as follows:
A = (s/2πr)πr^2 = (s/2r)r^2 = (s/2)r
Note that the area of a circular sector is proportional to the size of the central angle, and is equal to the fraction of the circle’s total area represented by the sector.
Arc length
The arc length of a circular sector is the length of the portion of the circumference of the circle that makes up the sector. The formula for the arc length of a circular sector is given by:
s = (θ/360) * 2πr
where θ is the central angle of the sector in degrees, π is the mathematical constant pi (approximately equal to 3.14), and r is the radius of the circle.
Alternatively, the formula can be written in terms of the area A of the sector, as follows:
s = 2r * (A / (πr^2)) = 2 * √(A / π)
Note that the arc length is proportional to the size of the central angle and the radius of the circle.
| A circular sector is a region of a circle enclosed between two radii and an arc of the circle. It is a portion of the circle and has the shape of a sector of a pie. The area of a circular sector can be found by multiplying the circle’s radius squared by the sector’s central angle (measured in radians), and dividing by 2. The formula is: A = (r^2 * θ)/2, where r is the radius of the circle and θ is the central angle of the sector in radians. | |
Area of the sector
| A circular sector is a region of a circle enclosed between two radii and an arc of the circle. It is a portion of the circle and has the shape of a sector of a pie. The area of a circular sector can be found by multiplying the circle’s radius squared by the sector’s central angle (measured in radians), and dividing by 2. The formula is: A = (r^2 * θ)/2, where r is the radius of the circle and θ is the central angle of the sector in radians. | |
Arc length
| A circular sector is a region of a circle enclosed between two radii and an arc of the circle. It is a portion of the circle and has the shape of a sector of a pie. The area of a circular sector can be found by multiplying the circle’s radius squared by the sector’s central angle (measured in radians), and dividing by 2. The formula is: A = (r^2 * θ)/2, where r is the radius of the circle and θ is the central angle of the sector in radians. | |
Interactive Graph
| From the interactive graph given below, drag the point ‘P’ to change the radius and, the points A or B to change the Angle | |