📝 Chapter Notes & Revision
Areas Related to Circles
📐 Formula & Cheat Sheet (English)
Quick Revision Notes & Formula Sheet
Class: 10th Mathematics
Chapter: Areas Related to Circles (वृत्तों से संबंधित क्षेत्रफल)
### Introduction & Basic Terms
- Circle (वृत्त): The locus of a point which moves in a plane in such a way that its distance from a fixed point is always constant. The fixed point is called the Center (केंद्र) and the constant distance is called the Radius (त्रिज्या - $r$).
- Diameter ($d$ - व्यास): A line segment passing through the center and whose endpoints lie on the circle. ($d = 2r$)
- Circumference (परिधि): The perimeter or boundary length of a circle. ($C = 2\pi r$)
- Area of Circle (वृत्त का क्षेत्रफल): $A = \pi r^2$
(Where $\pi \approx \frac{22}{7}$ or $3.14$)
### Important Formulas for Circle Components
| Term (पद) | Formula (सूत्र) | Description (विवरण) |
|---|---|---|
| Circumference (परिधि) | $2\pi r$ or $\pi d$ | Total length of the boundary |
| Area of Circle (वृत्त का क्षेत्रफल) | $\pi r^2$ | Space enclosed inside the circle |
| Area of Semicircle (अर्धवृत्त का क्षेत्रफल) | $\frac{1}{2}\pi r^2$ | Half of the circle's area |
| Perimeter of Semicircle (अर्धवृत्त का परिमाप) | $\pi r + 2r$ or $(\pi + 2)r$ | Boundary of semicircle including diameter |
| Area of Ring / Annulus (वलय का क्षेत्रफल) | $\pi(R^2 - r^2)$ | Area between two concentric circles of radii $R$ and $r$ ($R > r$) |
### Sector and Segment of a Circle (त्रिज्यखंड और वृत्तखंड)
Let $\theta$ be the angle of the sector (कोण) and $r$ be the radius of the circle (त्रिज्या).
1. Minor and Major Sector (लघु और दीर्घ त्रिज्यखंड)
- Area of Sector (त्रिज्यखंड का क्षेत्रफल): $$\text{Area} = \frac{\theta}{360^\circ} \times \pi r^2$$
- Length of an Arc of a Sector (चापि की लंबाई - $l$): $$l = \frac{\theta}{360^\circ} \times 2\pi r$$
- Relation between Area of Sector, Arc length, and Radius: $$\text{Area} = \frac{l \times r}{2}$$
- Area of Major Sector (दीर्घ त्रिज्यखंड का क्षेत्रफल): $$\text{Area of Circle} - \text{Area of Minor Sector} = \pi r^2 - \frac{\theta}{360^\circ} \times \pi r^2$$
2. Minor and Major Segment (लघु और दीर्घ वृत्तखंड)
- Area of Minor Segment (लघु वृत्तखंड का क्षेत्रफल): $$\text{Area of Minor Segment} = \text{Area of Minor Sector} - \text{Area of corresponding } \triangle OAB$$ $$\text{Area} = \left(\frac{\theta}{360^\circ} \times \pi r^2\right) - \left(\frac{1}{2} r^2 \sin\theta\right)$$
- Area of Major Segment (दीर्घ वृत्तखंड का क्षेत्रफल): $$\text{Area of Circle} - \text{Area of Minor Segment}$$
### Important Trigonometric Values for $\theta$ (Used in Sectors)
Keep these values handy when finding the area of segments where $\theta$ is given:
- $\sin(30^\circ) = \frac{1}{2}$, $\cos(30^\circ) = \frac{\sqrt{3}}{2}$
- $\sin(45^\circ) = \frac{1}{\sqrt{2}}$, $\cos(45^\circ) = \frac{1}{\sqrt{2}}$
- $\sin(60^\circ) = \frac{\sqrt{3}}{2}$, $\cos(60^\circ) = \frac{1}{2}$
- $\sin(90^\circ) = 1$, $\cos(90^\circ) = 0$
### Revolution of Wheels (चक्करों की संख्या)
If a wheel of radius $r$ rotates to cover a certain distance:
- Distance covered in 1 revolution (1 चक्कर में तय दूरी) = Circumference $= 2\pi r$
- Total Distance Covered (कुल तय दूरी) = Number of Revolutions $\times (2\pi r)$
- Number of Revolutions (चक्करों की संख्या) = $\frac{\text{Total Distance}}{\text{Circumference}}$
### Quick Tips for MP Board Exam Success
- Value of $\pi$: Always use $\pi = \frac{22}{7}$ unless specified otherwise in the question.
- Units: Pay careful attention to units. Area is always in square units ($cm^2, m^2$) and perimeter/length is in linear units ($cm, m$).
- Diagrams: Always draw a rough sketch for sector and segment problems to visualize the given data clearly.