This chapter explores the properties of circles, particularly focusing on tangents and their relationship with radii and secants.
Circles – Formula & Equation Sheet
Essential formulas and equations from Mathematics, tailored for Class X in Mathematics.
This one-pager compiles key formulas and equations from the Circles chapter of Mathematics. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
Length of tangent from external point (P) to circle: √(OP² - r²)
OP is the distance from the external point P to the center O of the circle, r is the radius. This formula calculates the length of the tangent from P to the circle.
Area of sector: (θ/360°) × πr²
θ is the central angle in degrees, r is the radius. This formula gives the area of a sector of a circle.
Length of arc: (θ/360°) × 2πr
θ is the central angle in degrees, r is the radius. This formula calculates the length of an arc of a circle.
Area of segment: Area of sector - Area of triangle
This formula finds the area of a segment by subtracting the area of the triangle from the area of the sector.
Circumference of circle: 2πr
r is the radius. This formula calculates the perimeter of a circle.
Area of circle: πr²
r is the radius. This formula finds the area enclosed by a circle.
Angle subtended by an arc at the center: 2 × angle subtended at any point on the remaining part of the circle
This theorem relates the angles subtended by an arc at the center and at any point on the circle.
Perpendicular from the center to a chord bisects the chord
This theorem states that if a perpendicular is drawn from the center of a circle to a chord, it divides the chord into two equal parts.
Equal chords are equidistant from the center
This theorem states that chords of equal length in a circle are equally distant from the center.
The tangent at any point of a circle is perpendicular to the radius through the point of contact
This theorem establishes the relationship between a tangent and the radius at the point of contact.
Equations
Theorem 10.1: The tangent at any point of a circle is perpendicular to the radius through the point of contact
This theorem is fundamental in understanding the properties of tangents to a circle.
Theorem 10.2: The lengths of tangents drawn from an external point to a circle are equal
This theorem is useful for solving problems involving tangents from an external point.
Equation of circle with center (h, k) and radius r: (x - h)² + (y - k)² = r²
This equation represents a circle in the coordinate plane.
Condition for a line y = mx + c to be tangent to the circle x² + y² = r²: c² = r²(1 + m²)
This condition checks if a line is tangent to a given circle.
Power of a point P(x₁, y₁) with respect to circle x² + y² + 2gx + 2fy + c = 0: x₁² + y₁² + 2gx₁ + 2fy₁ + c
This concept is used to find the relative position of a point with respect to a circle.
Angle between two tangents drawn from an external point to a circle: 180° - angle subtended by the line segment joining the points of contact at the center
This equation relates the angle between two tangents to the angle subtended at the center.
Length of the chord intercepted by a line y = mx + c on the circle x² + y² = r²: 2√(r² - c²/(1 + m²))
This formula calculates the length of the chord formed by the intersection of a line and a circle.
Condition for two circles to touch each other: Distance between centers = sum or difference of radii
This condition determines if two circles are tangent to each other.
Equation of tangent to the circle x² + y² = r² at point (x₁, y₁): xx₁ + yy₁ = r²
This equation gives the tangent to a circle at a specific point.
Equation of tangent to the circle x² + y² + 2gx + 2fy + c = 0 at point (x₁, y₁): xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = 0
This equation provides the tangent to a general circle at a given point.
This chapter introduces arithmetic progressions, which are sequences of numbers generated by adding a fixed value to the previous term. Understanding these patterns is crucial for solving real-life mathematical problems.
This chapter focuses on the properties of triangles, specifically their similarity and how it can be applied in various real-world contexts.
This chapter covers the concepts of coordinate geometry, including finding distances between points and dividing line segments. Understanding these concepts is essential for solving geometry problems using algebra.
This chapter focuses on the foundational concepts of trigonometry, particularly the relationships between the angles and sides of right triangles.
This chapter explores how trigonometry is applied in real-life situations, particularly in measuring heights and distances.
This chapter focuses on sectors and segments of circles, essential concepts in geometry. Understanding these helps in solving real-life problems related to areas and measurements.
This chapter explores how to find the surface areas and volumes of various solids, including combinations of basic shapes like cubes, cones, cylinders, and spheres, essential for real-world applications.
Statistics is the chapter that deals with the collection, analysis, interpretation, presentation, and organization of data.
This chapter explores the basic concepts and definitions of probability, highlighting its significance in predicting outcomes in uncertain situations.
