Circle Class 10 Cheatsheet

Cheatsheet Content

Introduction to Circles A circle is a collection of all points in a plane which are at a constant distance (radius) from a fixed point (center). Radius ($r$): The distance from the center to any point on the circle. Diameter ($d$): A line segment passing through the center and having its endpoints on the circle. $d = 2r$. Chord: A line segment joining any two points on a circle. Arc: A continuous piece of a circle. Minor Arc: Shorter arc. Major Arc: Longer arc. Sector: The region enclosed by two radii and an arc. Minor Sector: Enclosed by two radii and a minor arc. Major Sector: Enclosed by two radii and a major arc. Segment: The region enclosed by a chord and an arc. Minor Segment: Enclosed by a chord and a minor arc. Major Segment: Enclosed by a chord and a major arc. Terms Related to Circles Tangent: A line that intersects the circle at exactly one point. The point of intersection is called the point of contact . Secant: A line that intersects the circle at two distinct points. Concentric Circles: Circles with the same center but different radii. Theorems on Tangents Theorem 1: Radius and Tangent Perpendicularity The tangent at any point of a circle is perpendicular to the radius through the point of contact. If $AB$ is a tangent to the circle at point $P$ and $O$ is the center, then $OP \perp AB$. Theorem 2: Lengths of Tangents from an External Point The lengths of tangents drawn from an external point to a circle are equal. If $PA$ and $PB$ are two tangents drawn from an external point $P$ to a circle with center $O$, then $PA = PB$. Also, $OP$ bisects $\angle APB$ and $\angle AOB$. Properties of Tangents A tangent can be drawn to a circle at any point on it. There is only one tangent at any point on a circle. No tangent can be drawn to a circle from a point inside the circle. Two tangents can be drawn to a circle from a point outside the circle. The distance between two parallel tangents of a circle is equal to its diameter ($2r$). Formulas for Areas and Perimeters Circumference of a circle: $C = 2\pi r = \pi d$ Area of a circle: $A = \pi r^2$ Area of a sector with angle $\theta$ (in degrees): $A_{sector} = \frac{\theta}{360^\circ} \times \pi r^2$ Length of an arc with angle $\theta$ (in degrees): $L_{arc} = \frac{\theta}{360^\circ} \times 2\pi r$ Area of a segment: $A_{segment} = A_{sector} - A_{triangle}$ For a minor segment, $A_{segment} = \frac{\theta}{360^\circ} \pi r^2 - \frac{1}{2} r^2 \sin \theta$ Important Concepts & Relations The angle subtended by a chord at the center is $2\times$ the angle subtended by it at any point on the remaining part of the circle (for the same arc). Angles in the same segment of a circle are equal. The angle in a semicircle is a right angle ($90^\circ$). Cyclic Quadrilateral: A quadrilateral whose all four vertices lie on a circle. Sum of opposite angles is $180^\circ$. Summary of Key Relationships Geometric Figure Formula / Property Circle $C = 2\pi r$, $A = \pi r^2$ Sector (angle $\theta$) Area: $\frac{\theta}{360^\circ} \pi r^2$ Arc Length (angle $\theta$) Length: $\frac{\theta}{360^\circ} 2\pi r$ Tangent & Radius Perpendicular at point of contact Tangents from external point Lengths are equal ($PA=PB$)