Practice Problems on Surface Area and Volume of a Right Circular Cone

In this article, we learn about an important topic of class 9 Surface Area and Volume of a Right Circular Cone. This article will give details about the right circular cone. We will know how to find curved surface area, total surface area and volume of a cone.

Important Formulas of Right Circular Cone

Suppose, we have a right circular cone having,

• Base Radius ‘r’
• Slant Height ‘l’
• Height ‘h’

Then its various formulas are:

Curved Surface Area of Right Circular Cone	πrl
Total Surface Area of Right Circular Cone	πr(l+r)
Volume of Right Circular Cone	1/3 × πr2 × h

Practice Problems on Surface Area and Volume of a Right Circular Cone

Q1. Find the curved surface area of a cone, if its slant height is 48 cm and the radius of its base is 15 cm.

Q2. The radius of a cone is 8 cm and vertical height is 10 cm. Find the area of the curved surface.

Q3. The radius of a cone is 10 cm and area of curved surface is 314 cm². Find the slant height.

Q4. Find the total surface area of a right circular cone with radius 9 cm and height 12 cm.

Q5. Find the total surface area of a cone, if its slant height is 30 m and diameter of its base is 36 m.

Q6. Calculated total surface area of a cone is 90π cm². If the slant height of the cone be 10 cm, find the radius of the base.

Q7. Diameters of two cones are equal. If their slant heights are in the ratio 3:2, find the ratio of their curved surfaces.

Q8. Find the volume of the cone, if the radius is 7 cm and its height is 14 cm.

Q9. Volume of a given cone is 200π cubic centimeters. The given height of the cone is 15 cm. Find the radius of the cone.

Q10. There are two cones. Cone 1 has a radius of 6 cm and its height is 14 cm. Cone 2 has a radius of 10 cm and its height is 12 cm. Decide which cone has the greater volume.

Practice Problem on Surface Area and Volume of a Right Circular Cone with Solution

Problem 1: Find the curved surface area of a cone, if its slant height is 60 cm and the radius of its base is 21 cm.

Solution:

Given,

• Height = 60 cm
• Radius = 21 cm

Curved Surface Area of a Cone = πrl

= π × 21 × 60

= 3.14 × 1260

= 3956.4 cm²

Problem 2: Radius of a cone is 5 cm and vertical height is 12 cm. Find the area of the curved surface.

Solution:

Given,

• Height = 12 cm
• Radius = 5 cm

Curved Surface Area of a Cone = πrl

= π × 5 × 12

= 3.14 × 60

= 188.4 cm²

Problem 3: Radius of a cone is 7 cm and area of curved surface is 176 cm². Find the slant height.

Solution:

Given,

• Curved Surface Area = 176 cm²
• Radius = 5 cm

Curved Surface Area of a Cone = πrl

⇒ 176 = π × 5 × l

⇒ 176 = 15.7 × l

⇒ l = 11.21cm

So, height of the cone is 11.21 cm.

Problem 4: Find the total surface area of a right circular cone with radius 6 cm and height 8 cm.

Solution:

To calculate total surface area apply the following formula

Total Surface Area = πr(l+r)

• Radius = 6 cm
• Height = 8 cm

Total Surface Area = 3.14 × 6(8 + 6)

= 3.14 × 6 × 14

= 263.76 cm²

So, total surface area of the cone is 263.76 cm².

Problem 5: Find the total surface area of a cone, if its slant height is 21 cm and diameter of its base is 24 cm.

Solution:

To calculate total surface area apply the following formula

Total Surface Area = πr(l+r)

• Radius = 12cm
• Height = 8cm

Total Surface Area = 3.14 × 12(21 + 12)

= 3.14 × 12 × 33

= 1243.44 cm²

So, total surface area of the cone is 1243.44 cm².

Problem 6: Total surface area of a cone is 60π cm². If the slant height of the cone be 8 cm, find the radius of the base.

Solution:

• Slant Height = l = 4 cm
• Total Surface Area = 60π cm²

So, to find the radius

60π = πr(l+r)

60 = 4r + r²

r(r-6) + 10(r-6) = 0

(r-6)(r+10) = 0

r = 6 and r = -10(not possible)

So, radius of base is 6 cm.

Problem 7: Diameters of two cones are equal, the ratio of the slant height is 5:4, find the ratio of their curved surfaces.

Solution:

Curved Surface Area of Cone = πrl

Radius of First Cone(r₁) = Radius of Second Cone(r₂)

Slant Height of Cone(l₁):Slant Height of Cone(l₂) = 5:4

Required Ratio = πr₁l₁/πr₂l₂

= 5/4

So, ratio of curved surfaces is 5/4.

Problem 8: Find the volume of the cone, if the radius is 6cm and its height is 8cm.

Solution:

• Radius = 6 cm
• Height = 8 cm

Volume of Cone = 1/3 × (𝝅r²) × h

= 1/3 × 3.14 × 6 × 6 × 8

= 401.92 cm³

So, volume of the cone is 401.92 cm³.

Problem 9: The volume of a given cone is 100π cubic centimeters, The height of the cone is 10 cm. Find the radius of the cone.

Solution:

• Volume = 100 π
• Height = 10 cm

So, to find radius of cone

Volume = 1/3 × (𝝅r²) × h

100π = 1/3 × π × r × r × 10

r × r = 30

r = 5.4 cm

So, radius of cone is 5.4 cm.

Problem 10: There are two cones. Cone 1 has radius of 4cm and its height is 10cm. Cone 2 has radius of 6cm and its height is 8cm. So, decide which cone has the greater volume.

Solution:

To find volume of cone = 1/3 × (𝝅r²) × h

For Cone 1:

• Radius = 4 cm
• Height = 10 cm

Volume = 1/3 × 𝝅 × 4 × 4 × 10

Volume = 𝝅/3 × 160 cm³

For Cone 2:

• Radius = 6 cm
• Height = 8 cm

Volume = 1/3 × 𝝅 × 6 × 6 × 8

Volume = 𝝅/3 × 288 cm³

So, cone 2 has greater volume.

FAQs on Surface Area and Volume of a Right Circular Cone

If we have a right circular cone, then how many frustums does the cone can have?

In a right circular cone, we have ‘One’ frustums cone.

Find the curved surface area of right circular cone which have radius = 4cm and height = 6cm

Curved Surface Area = π×4×6 = 75.36 cm²

Write the formula for calculating the volume of right circular cone.

Formula for calculating the volume of right circular cone is Volume = 1/3 × πr²h

Can we find radius of cone if we have height and curved surface area value?

Yes, we can find the radius of cone if we have height and curved surface area value of the cone.