# Circle

## Important Formulae

Diameter of a circle is usually denoted by ‘D’.

Radius of a circle is denoted by ‘r’, thus, diameter = 2r

Circumference of circle = 2 π r, where r is the radius of circle

Area of circle = 2 π r^{2}

**Question 1:** Find the circumference of the circles with the following radius: (a) 14 cm (b) 28 mm (c) 21 cm

**Solution:** (a) Given, radius, r = 14 cm

We know that circumference `= 2 π r`

`= 2 xx (22)/(7)xx 14 = 2 xx 22 xx 2 = 88` cm

(b) Given, radius, r = 28 mm,

We know that circumference `= 2 π r`

`= 2 xx (22)/(7) xx 28 = 2 xx 22 xx 4 = 176` mm

(c) Given, radius, r = 21 cm

We know that circumference `= 2 π r`

`= 2 xx (22)/(7) xx 21 = 2 xx 22 xx 3 = 132` cm

**Question 2:** Find the area of the following circles, given that: (a) Radius = 14 cm, (b) Diameter = 49 mm, (c) Radius = 5 cm

**Solution:** (a) Given, radius, r = 14 cm

We know that area of a circle `= π r^2`

`= (22)/(7) xx 14^2`

`= (22)/(7) xx 14 xx 14`

`= 22 xx 2 xx 14 = 616` sq cm

(b) Given, diameter, D = 49 mm; or, radius = `(49)/(2)` mm

Area `= π r^2`

`= (22)/(7) xx (49)/(2) xx (49)/(2)`

`= 11 xx 7 xx (49)/(2)`

`= 11 xx 3.5 xx 49 = 1886.5 sq` mm

(c) Given, radius, r = 5 cm

Area `= π r^2`

`= (22)/(7) xx 5^2`

`= 3.14 xx 25 = 78.5` sq cm

**Question 3:** If the circumference of a circular sheet is 154 m, find its radius. Also find the area of the sheet.

**Solution:** Given, circumference = 154 m, radius = ?, Area = ?

Circumference of a circle `= 2 π r`

Therefore, `154 = 2 π r`

Or, `π r = (154)/(2) = 77`

Or, `r = (77)/(π) = 77 xx (7)/(22) = 24.5` m

Area `= π r^2`

`= (22)/(7) xx 24.5 xx 24.5`

`= 22 xx 3.5 xx 24.5 = 1886.5` sq m

**Question 3:** If the circumference of a circular sheet is 154 m, find its radius. Also find the area of the sheet.

**Answer:** Given, circumference = 154 m, radius = ?, Area = ?

Circumference of a circle `= 2 π r`

Therefore, `154 = 2 π r`

Or, `π r = (154)/(2) = 77`

Or, `r = (77)/(π) = 77 xx (7)/(22) = 24.5` m

Area `= π r^2`

`= (22)/(7) xx 24.5 xx 24.5`

`= 22 xx 3.5 xx 24.5 = 1886.5` sq m

**Question 4:** A gardener wants to fence a circular garden of diameter 21m. Find the length of the rope he needs to purchase, if he makes 2 rounds of fence. Also find the cost of the rope, if it costs Rs 4 per meter.

**Solution:** Given, Diameter = 21m, cost of the rope = Rs 4 per meter, length of the rope required to fence the garden two round=?, Cost of the rope = ?

The length of rope required `= 2 xx` circumference.

Circumference `= 2 π r = π D`

`= (22)/(7) xx 21 = 66` m

Therefore, length of rope required = circumference `xx 2 = 66 m xx 2 = 132` m

Cost of rope = Rate `xx` length

`= 4 xx 132 = Rs. 528`

**Question 5:** From a circular sheet of radius 4 cm, a circle of radius 3 cm is removed. Find the area of the remaining sheet. (Take π = 3.14)

**Solution:** Radius of circular sheet = 4cm,

Area `= π r^2`

`= 3.14 xx 4^2 = 3.14 xx 16= 50.24` cm^{2}

Radius of circular sheet which is removed = 3 cm

Area `= π r^2`

`= 3.14 xx 3^2 = 3.14 xx 9 = 28.26` cm^{2}

Now, Area of remaining sheet = Area of whole sheet — Area of removed sheet

= 50.24 cm^{2} — 28.26 cm^{2} = 21.98 cm^{2}

**Question 6:** Question:6. Saima wants to put a lace on the edge of a circular table cover of diameter 1.5m. Find the length of the lace required and also find its cost if one meter of the lace costs Rs 15 (Take π = 3.14)

**Solution:** Diameter of a circular table = 1.5 m, Cost of one metre of lace = Rs 15

Circumference of circle table `= π D`

`= 3.14 xx 1.5 = 4.71` m

Cost = Rate `xx` Length

`= Rs. 15 xx 4.71 = Rs. 70.65`