Arc and Sector Word Problem 1

Central angle of sector The length of an arc is equal to ratio of central angle(Ɵ) of the sector to central angle of the circle(3600) multiplied by the circumference of the circle. Central angle of sector Central angle of circle Length of the arc = x Circumference of circle Ɵ ∴ Length of the arc = The area of an arc is equal to ratio of central angle(Ɵ) of the sector to central angle of the circle(3600) multiplied by the area of the circle. Central angle of sector Central angle of circle Area of the arc = x area of circle Ɵ ∴ Area of the arc =

Perimeter of a sector What is perimeter? If we walk along the border of any shape, we get its perimeter. Perimeter of a sector Consider a sector of radius r and with the length of the arc l The perimeter of a sector is the sum of length of the arc and length of two radii. Perimeter of a sector = length of arc + length of 2 radii Perimeter of a sector = l + 2r ( where l = )

shaded with different colours. Find the length of each of the arcs. Example 1: A circular shaped gymnasium ring of radius 35cm is divided into 5 equal arcs shaded with different colours. Find the length of each of the arcs. Solution: Given: Radius of the ring = 35 cm , Ring is divided into 5 equal arcs To Find: Length of each arc We know that the length of the circle is the circumference of the circle The circle is divided into 5 arcs of equal length. The 5 arcs of equal length add up to the circumference of the circle. length of 5 arcs = circumference of circle length of 5 arcs = length of 1 arc = 5 = 220 cm = 44 cm Ans: Length of each arc = 44cm

Area of 6 sectors = Area of circle Example 2: A spinner of radius 7.5 cm is divided into 6 equal sectors. Find the area of each of the sectors. Solution: Given: Radius of the spinner = 7.5 cm ,Spinner is divided into 6 equal sectors To Find: Area of each sector The area of a sector is a part of the area of the circle. The circle (spinner) is divided into 6 equal sectors. The 6 sectors of equal area add up to the area of circle Area of 6 sectors = Area of circle Area of 6 sectors 11 2.5 Area of 1 sector 3 Ans: Area of each sector = 29.46 cm2

Given: Perimeter of the palm leaf fan( P)= 43 cm Example 3: Find the central angle and area of a palm leaf fan (sector) of radius 10.5 cm and whose perimeter is 43 cm. Solution: Given: Perimeter of the palm leaf fan( P)= 43 cm Radius of the palm leaf fan(r) = 10.5 cm To find: i) central angle ii) area of palm leaf fan To find the central angle of the sector, we need to find the length of the arc Length of the arc + 2 times radius of palm leaf fan = Perimeter of the palm leaf fan l + 2r = P l + 2 x r = P l + (2 x 10.5) = 43 (After substituting value of P=43 and r=10.5) l + 21 = 43 l = 43 – 21 = 22 cm l = 22 cm P = 43cm 10.5 cm

We found that the length of the arc (l) = 22 cm We know that = Length of the arc (l) 11 (substituting π = ,r=10.5 and l = 22) 180 90 60 2 Keeping Ɵ on the left side and transposing numbers to the right side, we get 1.5 Ɵ = 2 x 60 Ɵ = 120o Therefore, central angle (Ɵ) = 120o

Therefore, area of palm leaf fan = 115.5 cm2 1.5 3.5 3 = 22 x 1.5 x 3.5 = 115.5 cm2 Therefore, area of palm leaf fan = 115.5 cm2 Ans: i) Central angle Ɵ = 120o ii) Area of palm leaf fan = 115.5 cm2

Try These Q1) A circular wheel of radius 42cm is divided into 6 equal arcs. Find the length of each of the arcs. Q2) A dosa of radius 7 cm is divided into 3 equal sectors. Find the area of each of the sectors. Q3) Find the central angle and area of a pizza of radius 14 cm and whose perimeter is 39 cm