Circles – Formula & Equation Sheet
Essential formulas and equations from Mathematic, tailored for Class 10 in Mathematics.
This one-pager compiles key formulas and equations from the Circles chapter of Mathematic. 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
Circumference of a circle: C = 2πr
C represents the circumference (distance around the circle), r is the radius (distance from the center to the edge). This formula is used to calculate the total distance around a circle.
Area of a circle: A = πr²
A is the area (space inside the circle), r is the radius. This formula is essential for finding the space that a circle occupies.
Length of an arc: L = (θ/360) × 2πr
L is the length of the arc, θ is the central angle in degrees. This formula calculates the distance along the curved part of the circle for a given angle.
Area of a sector: A = (θ/360) × πr²
A is the area of the sector, θ is the central angle in degrees. This formula allows computation of the area of a pie-slice portion of the circle.
Tangent-secant theorem: PT² = OP × OQ
PT is the tangent length, OP and OQ are distances from the circle's center to the points where the secant intersects the circle. This theorem is useful in problems involving tangents and secants.
Theorem 10.1: Tangent to a circle is perpendicular to the radius.
At any point of tangency P, the tangent line is perpendicular to the radius OP. This property is crucial for solving many geometric problems related to circles.
Theorem 10.2: Lengths of tangents from an external point are equal.
If two tangents are drawn from a point P to a circle, then their lengths (PQ and PR) are equal. This is useful for determining tangent lengths in geometric constructions.
Length of tangent from a point to a circle: L = √(d² - r²)
L is the length of the tangent, d is the distance from the external point to the center of the circle, r is the radius. This formula calculates the tangent length when the distance from the point to the center and the radius are known.
Angle between two tangents from an external point: ∠PTQ = 180° - 2∠POQ
PT and TQ are tangents from point P to the circle, O is the center. This relationship helps in calculating angles in tangent problems.
Property of chords: If a chord is perpendicular to a radius, it bisects the chord.
This property states that if a radius is drawn perpendicular to a chord, it will divide the chord into equal parts. This is essential for many circle proofs.
Equations
d = √(r² + L²)
d is the distance from the center of the circle to a point on the tangent, r is the radius, and L is the length of the tangent. This is used to find the distance from the circle’s center to a tangent point.
PQ = √(OP² - OQ²)
PQ is the length of the tangent, OP is the distance from the external point to the center, OQ is the radius. This equation can be used to find lengths in scenarios involving tangents.
AB + CD = AD + BC (for tangential quadrilaterals)
This relationship between the sides of a quadrilateral circumscribing a circle helps ascertain the equality of opposite sides, useful in various geometry problems.
OP = r (for the tangent at point P)
OP is the radius to point P, r is the radius length. This emphasizes the nature of tangents as they relate to circle radii.
Area of inscribed triangle: A = ½ × a × r
A is the area of the triangle, a is the length of the base, r is the radius of the inscribed circle (incircle). This formula is valuable for solving problems involving triangles and inscribed circles.
∠OQP + ∠QRP = 90°
In triangle OQP, OQ is perpendicular to PQ. Understanding this helps in solving angle-related geometry problems.
a + b = diameter (for circles)
Here, a and b represent the lengths of two line segments such that together they equal the diameter of the circle. This is often used in laying out structures.
AP² = AO² - OP²
AP is a segment from point A to point P on the circle, AO is the distance from A to the center O. This equation is another aspect of working with tangents and segments.
PQ² + r² = OP²
This relates the square of the tangent length (PQ), the circle's radius (r), and the distance to the external point (OP). Useful for proofs and cone problems.
[Equation of Circle]: (x - h)² + (y - k)² = r²
This represents a circle with center (h, k) and radius r in the Cartesian plane. Understanding its equation is fundamental in analytical geometry.
