Cylinder, Cone and Sphere Chapter Notes | Mathematics Class 10 ICSE PDF Download

Introduction

Imagine shaping clay into a tall water tank, a pointy ice-cream cone, or a perfectly round football. These shapes—cylinders, cones, and spheres—are not just fascinating to look at but are also all around us in everyday life. From water pipes to party hats, understanding their surface areas and volumes helps us solve real-world problems, like calculating how much paint is needed for a cylindrical drum or how much ice cream fits in a conical cup. This chapter takes you on an exciting journey through these 3D shapes, unraveling their properties with simple formulas and practical examples. Let’s dive into the world of mensuration and explore how to measure these solids!

Cylinder

• A cylinder is a solid with a uniform circular cross-section, like a can or a pipe.
• It has two parallel circular bases connected by a curved surface.

Formulas:

• Area of cross-section = πr²
• Circumference of cross-section = 2πr
• Curved surface area = 2πrh
• Total surface area = 2πrh + 2πr² = 2πr(h + r)
• Volume = πr²h

Example: The area of the curved surface of a cylinder is 4,400 cm² and the circumference of its base is 110 cm. Find the height and volume.

• Step 1: Curved surface area = 2πrh = 4,400 cm², and circumference = 2πr = 110 cm.
• Step 2: Divide the curved surface area by circumference: h = 4,400 ÷ 110 = 40 cm.
• Step 3: Find radius: 2πr = 110, so r = 110 ÷ (2 × 22/7) = 35/2 cm.
• Step 4: Volume = πr²h = (22/7) × (35/2)² × 40 = 38,500 cm³.

Answer: Height = 40 cm, Volume = 38,500 cm³.

Hollow Cylinder

Hollow Cylinder Volume

• A hollow cylinder has an inner and outer circular surface, like a pipe or a straw.
• It has an external radius (R), internal radius (r), and height (h).
• Thickness of wall = R - r
• Area of cross-section = π(R² - r²)
• External curved surface area = 2πRh
• Internal curved surface area = 2πrh
• Total surface area = 2πRh + 2πrh + 2π(R² - r²)
• Volume of material = π(R² - r²)h
• Determine the external radius (R), internal radius (r), and height (h).
• Calculate the thickness by subtracting r from R.
• Find the cross-sectional area by subtracting the inner circle area from the outer.
• Compute external and internal curved surfaces using their respective radii.
• For total surface area, sum both curved surfaces and the area of the annular bases.
• For volume, multiply the cross-sectional area by the height.

Example:. Cylindrical tube, open at both ends, has an internal diameter of 11.2 cm, length 21 cm, and metal thickness of 0.4 cm. Calculate the volume of the metal.

• Step 1: Internal radius (r) = 11.2 ÷ 2 = 5.6 cm.
• Step 2: External radius (R) = 5.6 + 0.4 = 6.0 cm.
• Step 3: Volume of metal = π(R² - r²)h = (22/7) × (6² - 5.6²) × 21.
• Step 4: Calculate: (6² - 5.6²) = 36 - 31.36 = 4.64, so volume = (22/7) × 4.64 × 21 = 306.24 cm³ ≈ 306.2 cm³.

Answer: Volume = 306.2 cm³.

Cone

• A cone is formed by revolving a right-angled triangle around one of its sides (not the hypotenuse), like a party hat.
• It has a circular base (radius r), height (h), and slant height (l).
• Slant height: l² = h² + r²
• Volume = (1/3)πr²h
• Curved surface area = πrl
• Total surface area = πrl + πr² = πr(l + r)
• Identify the radius (r), height (h), and find slant height (l) using the Pythagorean theorem.
• Calculate volume by multiplying the base area by height and dividing by 3.
• Find curved surface area by multiplying π, radius, and slant height.
• Add the base area to the curved surface area for total surface area.

Example:A cone has a base radius of 7 cm and height of 24 cm. Find its volume and total surface area.

• Step 1: Slant height: l = √(24² + 7²) = √(576 + 49) = 25 cm.
• Step 2: Volume = (1/3) × (22/7) × 7² × 24 = 1,232 cm³.
• Step 3: Total surface area = πr(l + r) = (22/7) × 7 × (25 + 7) = 704 cm².

Answer: Volume = 1,232 cm³, Total surface area = 704 cm².

Sphere

• A sphere is formed by revolving a circle around its diameter, like a ball.
• It has a radius (r).
• Volume = (4/3)πr³
• Surface area = 4πr²
• Identify the radius (r).
• For volume, multiply (4/3) by π and r³.
• For surface area, multiply 4 by π and r².

Example:. sphere’s surface area is 616 cm². Find its volume.

• Step 1: Surface area = 4πr² = 616, so r² = 616 × 7 ÷ (4 × 22) = 49, r = 7 cm.
• Step 2: Volume = (4/3) × (22/7) × 7³ = 1,437 (1/3) cm³.

Answer: Volume = 1,437 (1/3) cm³.

Spherical Shell

Spherical Shell Volume

• A spherical shell is the solid between two concentric spheres, like a hollow ball.
• It has an external radius (R) and internal radius (r).

Formula:

• Volume = (4/3)π(R³ - r³)
• Identify the external (R) and internal (r) radii.
• Calculate the volume by subtracting the inner sphere’s volume from the outer sphere’s volume.

Example: A hollow sphere with internal and external diameters 4 cm and 8 cm is melted into a cone with base diameter 8 cm. Find the cone’s height.

• Step 1: Internal radius (r) = 2 cm, external radius (R) = 4 cm.
• Step 2: Volume of spherical shell = (4/3)π(4³ - 2³) = (4/3)π × 56.
• Step 3: Cone’s radius = 4 cm, volume = (1/3)π × 4² × h.
• Step 4: Equate volumes: (1/3)π × 16h = (4/3)π × 56, so h = 14 cm.

Answer: Height = 14 cm.

Hemisphere

• A hemisphere is half a sphere, formed by cutting a sphere through its center.
• Volume = (2/3)πr³
• Total surface area = 3πr²
• For volume, use half the sphere’s volume formula.
• For total surface area, add the curved surface (2πr²) and the base area (πr²).

Example:A hollow hemispherical vessel has internal and external diameters of 42 cm and 45.5 cm. Find its capacity and outer curved surface area.

• Step 1: Internal radius (r) = 21 cm, external radius (R) = 22.75 cm.
• Step 2: Capacity = (2/3)πr³ = (2/3) × (22/7) × 21³ = 19,404 cm³.
• Step 3: Outer curved surface area = 2πR² = 2 × (22/7) × 22.75² = 3,253.25 cm².

Answer: Capacity = 19,404 cm³, Outer curved surface area = 3,253.25 cm².

Conversion of Solids

Sphere Conversion

Conversion involves melting one solid and recasting it into another, keeping the volume constant.

• Calculate the volume of the original solid.
• Equate it to the volume of the new solid to find unknown dimensions.
• Solve for the required dimension or number of new solids formed.

Example:. sphere of radius 10.5 cm is melted and recast into small cones of radius 3.5 cm and height 3 cm. Find the number of cones.

• Step 1: Volume of sphere = (4/3)π × 10.5³.
• Step 2: Volume of one cone = (1/3)π × 3.5² × 3.
• Step 3: Number of cones = [(4/3)π × 10.5³] ÷ [(1/3)π × 3.5² × 3] = 126.

Answer: Number of cones = 126.

Combination of Solids

Combination Solids

Combination solids are made by joining two or more solids, like a cone on a hemisphere.

• Calculate the volume or surface area of each solid separately.
• For volume, add the volumes of all solids.
• For surface area, add the exposed surfaces, subtracting any overlapping areas.

Example:. toy is a cone mounted on a hemisphere, both with a radius of 6 cm. The cone’s height is 8 cm. Find the surface area and volume (π = 3.14).

• Step 1: Slant height of cone: l = √(8² + 6²) = 10 cm.
• Step 2: Surface area = πrl + 2πr² = 3.14 × (6 × 10 + 2 × 6²) = 414.48 cm².
• Step 3: Volume = (1/3)πr²h + (2/3)πr³ = 3.14 × [(1/3) × 6² × 8 + (2/3) × 6³] = 753.6 cm³.

Answer: Surface area = 414.48 cm², Volume = 753.6 cm³.

Miscellaneous Problems

Cylindrical Solid

These involve complex solids with cavities or combinations of shapes.

• Identify the solids and their dimensions.
• Calculate the volume or surface area of the main solid.
• Subtract the volume or adjust the surface area for any cavities or added parts.

Example: A cylinder (height 36 cm, radius 14 cm) has a conical cavity (radius 7 cm, height 24 cm) drilled out. Find the volume and total surface area of the remaining solid.

• Step 1: Cylinder volume = π × 14² × 36 = (22/7) × 196 × 36 = 22,176 cm³.
• Step 2: Cone volume = (1/3)π × 7² × 24 = (1/3) × (22/7) × 49 × 24 = 1,232 cm³.
• Step 3: Remaining volume = 22,176 - 1,232 = 20,944 cm³.
• Step 4: Cone’s slant height: l = √(24² + 7²) = 25 cm.
• Step 5: Surface area = πR² + 2πRH + π(R² - r²) + πrl = (22/7) × [14² + 2 × 14 × 36 + (14² - 7²) + 7 × 25] = 4,796 cm².

Answer: Volume = 20,944 cm³, Total surface area = 4,796 cm².