Introduction to Sectors
A sector is a portion of a circle enclosed by two radii and the arc between them. If you were to draw two radii from the centre of a circle to its boundary, the space enclosed between these radii and the arc connecting their endpoints would form a sector. For related concepts, you might find our notes on arc length useful.
Deriving the Formula
To understand the formula for the area of a sector, it's essential to first grasp the relationship between the whole circle and its parts. The area of a full circle is given by πr2, where r is the radius of the circle. Now, consider a sector that subtends an angle θ (in radians) at the centre. The proportion of the circle that this sector represents is θ/2π (since there are 2π radians in a full circle).
Thus, the area of the sector is a fraction of the area of the whole circle. This fraction is given by the ratio of the angle subtended by the sector to the total angle in a circle (which is 2π radians).
Mathematically, this relationship can be expressed as: Sector Area (A) = (θ/2π) x πr2 Simplifying, we get: A = 0.5 x r2 x θ
Where:
- A is the sector area.
- r is the radius of the circle.
- θ is the angle in radians subtended by the arc at the centre of the circle. This formula can be linked to other areas of mathematics, such as quadratic functions and their applications.
Applications of Sector Area
The concept of sector area is not just a theoretical one; it has numerous practical applications:
1. Cartography and Navigation: When charting paths on maps, especially on global scales, the sector area helps in determining the actual area covered by a specific path on the Earth's surface. This can be related to understanding correlation coefficients in data analysis.
2. Agriculture: Farmers often use the concept when planning irrigation for circular fields or when designing circular crop patterns.
3. Urban Planning: City planners use the sector area concept when designing roundabouts, circular parks, and other circular structures.
4. Fashion and Design: The concept is used in designing jewellery, especially those with circular patterns, and in fashion, when designing dresses with flared circular patterns.
5. Education: Teachers use sectors to explain fractions and percentages visually. For instance, a sector representing 1/4th of a circle can be used to explain 25%. This can also be extended to topics like binomial distribution.
In-depth Examples
Example 1: A circular park has a radius of 10 metres. A section of the park is designated for a flower show, forming a sector with an angle of 2 radians at the centre. What is the area of this section?
Solution: Using the formula: A = 0.5 x r2 x θ A = 0.5 x 102 x 2 A = 0.5 x 100 x 2 A = 100 square metres The area designated for the flower show is 100 square metres.
Example 2: A fan is designed in the shape of a sector. If the fan has a radius of 20 cm and covers an angle of 1.5 radians when fully opened, what is the area covered by the fan?
Solution: Using the formula: A = 0.5 x 202 x 1.5 A = 0.5 x 400 x 1.5 A = 300 square centimetres The area covered by the fan when fully opened is 300 square centimetres. For further mathematical understanding, you can also refer to our notes on exponential equations.
FAQ
Yes, the formula for sector area can be used for any sector, regardless of its size. Whether the sector is less than, equal to, or greater than half a circle, the formula A = 0.5 x r2 x theta remains valid. The angle theta in radians will simply reflect the size of the sector. For instance, for three-quarters of a circle, theta would be 1.5pi radians.
The sector area is directly proportional to the square of the radius. If the radius is doubled, the sector area will increase by a factor of four. This is because the formula for sector area is A = 0.5 x r2 x theta. If r becomes 2r, the new area A' will be A' = 0.5 x (2r)2 x theta = 4 x 0.5 x r2 x theta, which is four times the original area.
The formula A = 0.5 x r2 x theta is specific to circles. Ellipses, while similar in shape to circles, have two distinct axes (major and minor), and their area is calculated differently. The sector of an ellipse would also have a different formula for its area, derived from the general area formula for ellipses. For circles, the simplicity arises because all radii are of equal length, which isn't the case for ellipses.
Radians are a natural way to measure angles in terms of the radius of a circle. When we use radians, the formula for the sector area becomes more streamlined and directly relates the angle to the radius of the circle. Using degrees would require an additional conversion factor, complicating the formula. Moreover, in advanced maths and calculus, functions and formulas often have simpler and more elegant forms when expressed in radians, making calculations more straightforward.
The formula for the area of a circle is given by A = pi r2, where r is the radius of the circle. When we consider a sector of the circle, we're essentially looking at a fraction of the entire circle. This fraction is determined by the angle the sector subtends at the centre, in relation to the total angle in a circle (which is 360° or 2pi radians). If a sector subtends an angle theta at the centre, the fraction of the circle it represents is theta/360 (in degrees) or theta/2pi (in radians). Thus, the area of the sector is this fraction multiplied by the area of the entire circle, leading to the formula A = 0.5 x r2 x theta (in radians).
Practice Questions
To find the area of the sector, we first need to convert the angle from degrees to radians. The conversion is given by: θ (in radians) = θ (in degrees) × (π/180) θ = 100° × (π/180) = (5π/9) radians. Now, using the formula for the sector area: A = 0.5 × r2 × θ A = 0.5 × 122 × (5π/9) A = 0.5 × 144 × (5π/9) A = 80π/3 or approximately 83.78 square metres. The area of the section of the pond is approximately 83.78 square metres.
A semi-circular brooch implies that the brooch is half a circle. A quarter section of this would mean 1/4 of a semi-circle, which is 1/8 of a full circle. The angle subtended by 1/8 of a circle at the centre is (360°/8) = 45°. First, we convert this angle to radians: θ = 45° × (π/180) = π/4 radians. Now, using the formula for the sector area: A = 0.5 × r2 × θ A = 0.5 × 52 × π/4 A = 0.5 × 25 × π/4 A = 25π/8 or approximately 9.82 square centimetres. The area where the gemstones will be embedded is approximately 9.82 square centimetres.
