Solution: In the circle with radius r and the angle at the centre with a degree measure of θ, (i) Length of the Arc = θ/360° × 2πr (ii) Area of the sector = θ/360° × πr2 (iii) Area of the segment = Area of the sector - Area of the corresponding triangle Let's draw a figure to visualize the problem. Here, r = 21 cm, θ = 60° Visually it’s clear from the figure that, Area of the segment APB = Area of sector AOPB - Area of ΔAOB (i) Length of the Arc, APB = θ/360° × 2πr = 60°/360° × 2 × 22/7 × 21 cm = 22 cm (ii) Area of the sector, AOBP = θ/360° x πr2 = 60°/360° × 22/7 × 21 × 21 cm2 = 231 cm2 (iii) Area of the segment = Area of the sector AOBP - Area of the triangle AOB To find the area of the segment, we need to find the area of ΔAOB In ΔAOB, draw OM ⊥ AB. Consider ΔOAM and ΔOMB, OA = OB (radii of the circle) OM = OM (common side) ∠OMA = ∠OMB = 90° (Since OM ⊥ AB) Therefore, ΔOMB ≅ ΔOMA (By RHS Congruency) So, AM = MB (Corresponding parts of the congruent triangles are always equal) ∠AOM = ∠BOM = 1/2 × 60° = 30° In ΔAOM, cos 30° = OM/OA and sin 30° = AM/OA √3/2 = OM/r and 1/2 = AM/r OM = (√3/2) r and AM = (1/2) r AB = 2AM AB = 2 × (1/2) r AB = r Therefore, area of ΔAOB = 1/2 × AB × OM = 1/2 × r × (√3/2) r = 1/2 × 21 cm × (√3/2) × 21 cm = 441√3/4 cm2 Area of the segment formed by the chord = Area of the sector AOBP - Area of the triangle AOB = (231 - 441√3/4) cm2 ☛ Check: NCERT Solutions Class 10 Maths Chapter 12 Video Solution: NCERT Solutions Class 10 Maths Chapter 12 Exercise 12.2 Question 5 Summary: The length of the arc APB, area of the sector AOBP and area of the segment of a circle of radius 21 cm in which an arc subtends an angle of 60° at the centre are 22 cm, 231 cm2 and (231 - 441√3/4) cm2 respectively. ☛ Related Questions: This arc length calculator is a tool that can calculate the length of an arc and the area of a circle sector. This article explains the arc length formula in detail and provides you with step-by-step instructions on how to find the arc length. You will also learn the equation for sector area. In case you're new to circles, calculating the length and area of sectors could be a little advanced, and you need to start with simpler tools, such as circle length and circumference and area of a circle calculators.
The length of an arc depends on the radius of a circle and the central angle θ. We know that for the angle equal to 360 degrees (2π), the arc length is equal to circumference. Hence, as the proportion between angle and arc length is constant, we can say that: L / θ = C / 2πAs circumference C = 2πr, L / θ = 2πr / 2π L / θ = rWe find out the arc length formula when multiplying this equation by θ: L = r * θHence, the arc length is equal to radius multiplied by the central angle (in radians).
We can find the area of a sector of a circle in a similar manner. We know that the area of the whole circle is equal to πr². From the proportions, A / θ = πr² / 2π A / θ = r² / 2The formula for the area of a sector is: A = r² * θ / 2
Make sure to check out the equation of a circle calculator, too!
To calculate arc length without radius, you need the central angle and the sector area:
Or the central angle and the chord length:
To calculate arc length without the angle, you need the radius and the sector area:
Or you can use the radius and chord length:
Arc length is a measurement of distance, so it cannot be in radians. The central angle, however, does not have to be in radians. It can be in any unit for angles you like, from degrees to arcsecs. Using radians, however, is much easier for calculations regarding arc length, as finding it is as easy as multiplying the angle by the radius. |