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.
Areas Related to 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 Areas Related to 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
Area of a circle: A = πr²
A represents the area of the circle, π is a constant (≈ 3.14 or 22/7), and r is the radius of the circle. This formula calculates the space inside the circle.
Circumference of a circle: C = 2πr
C is the circumference, π is pi, and r is the radius. It measures the perimeter of the circle.
Area of a sector: A = (θ/360) × πr²
A is the area of the sector, θ is the central angle in degrees, π is pi, and r is the radius. It calculates the area of a pie-shaped part of the circle.
Length of an arc: L = (θ/360) × 2πr
L is the arc length, θ is the central angle, π is pi, and r is the radius. This finds the length of the curved part of the sector.
Area of a segment: A = (θ/360) × πr² - (1/2)r²sinθ
A is the segment area, θ is the central angle, π is pi, and r is the radius. It calculates the area between a chord and its arc.
Area of major sector: A = πr² - (θ/360) × πr²
A is the area of the major sector, θ is the minor sector's angle, π is pi, and r is the radius. It finds the larger area outside the minor sector.
Area of major segment: A = πr² - [(θ/360) × πr² - (1/2)r²sinθ]
A is the area of the major segment, θ is the central angle of the minor segment, π is pi, and r is the radius. It calculates the larger area outside the minor segment.
Perimeter of a sector: P = 2r + (θ/360) × 2πr
P is the perimeter, r is the radius, θ is the central angle, and π is pi. It sums the arc length and the two radii.
Area of a quadrant: A = (1/4)πr²
A is the area of the quadrant, π is pi, and r is the radius. A quadrant is a sector with a 90-degree angle.
Perimeter of a quadrant: P = 2r + (1/4) × 2πr
P is the perimeter, r is the radius, and π is pi. It includes two radii and a quarter of the circumference.
Equations
Relation between sector area and arc length: A = (L × r)/2
A is the sector area, L is the arc length, and r is the radius. This connects the area of a sector with its arc length.
Central angle from arc length: θ = (L × 360)/(2πr)
θ is the central angle in degrees, L is the arc length, π is pi, and r is the radius. It finds the angle subtended by an arc.
Radius from sector area: r = √[(A × 360)/(θ × π)]
r is the radius, A is the sector area, θ is the central angle, and π is pi. It derives the radius when the area and angle are known.
Chord length from central angle: c = 2r sin(θ/2)
c is the chord length, r is the radius, and θ is the central angle. It calculates the straight line connecting two points on the circle.
Area of an equilateral triangle inscribed in a circle: A = (3√3/4)r²
A is the area, r is the radius of the circumscribed circle. It's useful for problems involving circles and inscribed triangles.
Angle subtended by a chord at the center: θ = 2 arcsin(c/2r)
θ is the central angle, c is the chord length, and r is the radius. It finds the angle based on the chord.
Area of a circular ring: A = π(R² - r²)
A is the area of the ring, R is the outer radius, r is the inner radius, and π is pi. It calculates the area between two concentric circles.
Perimeter of a semicircle: P = πr + 2r
P is the perimeter, r is the radius, and π is pi. It includes half the circumference and the diameter.
Area of a semicircle: A = (1/2)πr²
A is the area, r is the radius, and π is pi. It calculates half the area of a full circle.
Relation between area and circumference: A = C²/(4π)
A is the area, C is the circumference, and π is pi. It connects the area of a circle directly with its circumference.
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