Area and Arc length
Area and Arc Length
Area of Circle
- The area of a circle is given by the formula A = πr², where r represents the radius of the circle.
- This is derived from the concept of breaking a circle up into an infinite number of tiny triangles with base along the circumference and height equal to the radius.
Area of Sector
- A sector of a circle is like a ‘slice’ of the circle.
- The area of a sector is given by the formula A = 1/2*r²θ, where r is the radius and θ is the angle at the centre of the circle in radians.
- If the angle is given in degrees, the formula becomes A = (θ/360) * πr², where θ is the angle at the centre in degrees.
- This is derived from considering the fraction of the total area of the circle which the sector represents, determined by the fraction of the full 360 degree or 2π radian angle which the sector spans.
Arc Length
- The length of an arc of a circle is calculated using the formula L = rθ, where r is the radius and θ is the angle at the centre of the circle in radians.
- If the angle is given in degrees, the formula becomes L = (θ/360) *2πr, where θ is the angle at the centre in degrees.
- It represents the distance you would travel if you went around the circumference of the circle from one end of the arc to the other.
Practical Applications
- Understanding how to find the area of a circle, the area of a sector, and the arc length has varoius real-life applications, such as surveying, navigation, art, and architecture.
- These formulas are also foundational for further studies in calculus and physics.
Revise the key concepts related to the area and arc length, and keep working on practice problems related to these topics to reinforce your understanding. Remember to check whether angles are given in degrees or radians and use the appropriate formula accordingly.