Explore the properties, theorems, and applications of circles in geometry, including tangents, chords, and angles subtended by arcs.
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.
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