NCERT Class X Chapter 11: Area Related To Circles Exercise 11.1 Question 9

NCERT Class X Chapter 11: Area Related To Circles

Question:

A brooch is made with silver wire in the form of a circle with diameter 35 mm. The wire is also used in making 5 diameters which divide the circle into 10 equal sectors. Find:
(i) the total length of the silver wire required.
(ii) the area of each sector of the brooch.
(Use \( \pi = 3.14 \))

Given:

  • Diameter of the circle, \( d = 35 \) mm
  • Number of diameters used = 5
  • Number of sectors = 10
  • \( \pi = 3.14 \)

To Find:

  • (i) Total length of silver wire required
  • (ii) Area of each sector of the brooch

Formula:

  • Circumference of a circle:   \( C = \pi d \)
  • Area of a circle:   \( A = \pi r^2 \)
  • Area of each sector:   \( \text{Area of sector} = \dfrac{\text{Area of circle}}{\text{Number of sectors}} \)

Solution:

Step 1: Find the circumference of the circle.

$$ \text{Circumference} = \pi d = 3.14 \times 35 = 109.9 \ \text{mm} $$

Step 2: Calculate the total length of the 5 diameters.

$$ \text{Length of 5 diameters} = 5 \times 35 = 175 \ \text{mm} $$

Step 3: Add the circumference and the length of 5 diameters to get the total length of silver wire required.

$$ \text{Total length of wire} = 109.9 + 175 = 284.9 \ \text{mm} $$

Step 4: Find the radius of the circle.

$$ r = \frac{d}{2} = \frac{35}{2} = 17.5 \ \text{mm} $$

Step 5: Calculate the area of the circle.

$$ \text{Area of circle} = \pi r^2 = 3.14 \times (17.5)^2 = 3.14 \times 306.25 = 961.625 \ \text{mm}^2 $$

Step 6: Find the area of each sector by dividing the area of the circle by the number of sectors.

$$ \text{Area of each sector} = \frac{961.625}{10} = 96.1625 \ \text{mm}^2 $$

Result:

  • (i) Total length of silver wire required = 284.9 mm
  • (ii) Area of each sector of the brooch = 96.1625 mm2
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