Surface Areas and Volumes (Class 10): Notes और MCQ Practice

आसान भाषा में

इस chapter में हम basic 3D shapes (Cylinder, Cone, Sphere, Hemisphere) को जोड़कर बने नए solids का Surface Area और Volume निकालना सीखते हैं। Simple फंडा यह है कि जब shapes को combine करते हैं, तो Volume सीधे add होता है, लेकिन Surface Area में केवल वही surface गिना जाता है जो बाहर से touch किया जा सके!

Important Formulas & Points

• Cone:
• Slant height ($l$) = √(r² + h²)
• Curved Surface Area (CSA) = πrl
• Total Surface Area (TSA) = πr(l + r)
• Volume = 1/3 × πr²h
• Cylinder: CSA = 2πrh | TSA = 2πr(h + r) | Volume = πr²h
• Sphere: Surface Area = 4πr² | Volume = 4/3 × πr³
• Hemisphere: CSA = 2πr² | TSA = 3πr² | Volume = 2/3 × πr³
• Conversion of Solids: जब किसी shape को melt करके नया shape बनाया जाता है, तो दोनों का Volume हमेशा same रहता है।

2 Memory Tricks

1. Cone vs Cylinder: सेम radius और height पर, Cone की Volume हमेशा Cylinder की Volume की 1/3 होती है (Volume = 1/3 × πr²h)।
2. Hemisphere Base Logic: Curved हिस्सा = 2πr²। TSA के लिए flat circular base (πr²) जोड़ दो, बन जाएगा 3πr²।

2 Common Board-Exam Mistakes

1. Direct TSA जोड़ना: दो shapes को मिलाकर बने solid का TSA निकालते समय बच्चे दोनों का TSA जोड़ देते हैं। Hidden/joined surfaces को हमेशा छोड़ना होता है (जैसे Cylinder + Cone में base area छिप जाता है)।
2. 'h' और 'l' का Confusion: Cone के CSA (πrl) में slant height $l$ की जगह गलती से vertical height $h$ use कर देना। हमेशा check करें कि question में $l$ दिया है या $h$!

Surface Areas and Volumes - Class 10 Study Notes

1. Chapter in Short

In earlier classes, you learned how to find the surface area and volume of single 3D shapes like cubes, cuboids, cylinders, cones, and spheres.

In Class 10, the focus is on combinations of solids—objects formed by joining two or more basic 3D shapes together (for example, an ice-cream cone, a circus tent, or a medicine capsule). The chapter teaches you how to find:

• Total Surface Area (TSA) of combined solids.
• Volume of combined solids.

2. Key Concepts Explained Simply

A. Surface Area of Combined Solids

When two solids are joined together, the surface area of the new shape is not simply the sum of the total surface areas of the individual shapes.

• Why? Because the faces where the two solids touch each other get glued together and are hidden inside. Surface area only measures the outer visible skin that you can touch or paint.
• Golden Rule:

Total Surface Area of New Solid = Visible Curved Surface Area (CSA) of Solid 1 + Visible Curved Surface Area (CSA) of Solid 2

• Everyday Example: Think of an ice cream cone with a scoop of ice cream on top. It is a cone joined to a hemisphere. When you wrap it, you only cover the curved side of the cone and the curved dome of the hemisphere. You do not cover the circular flat top of the cone or the circular flat bottom of the scoop because they are hidden inside.

B. Volume of Combined Solids

Unlike surface area, volume is straightforward because space inside simply adds up.

• Golden Rule:

Total Volume of New Solid = Volume of Solid 1 + Volume of Solid 2

• If a hollow cavity is carved out of a solid:

Remaining Volume = Volume of Original Solid − Volume of the Cavity Carved Out

3. Important Formulas and Key Facts

(Take π = 22/7 unless specified as 3.14)

1. Cuboid

• Curved/Lateral Surface Area (Area of 4 walls) = 2(l + b)h
• Total Surface Area (TSA) = 2(lb + bh + hl)
• Volume = l × b × h

2. Cube (Edge = a)

• Lateral Surface Area = 4a²
• Total Surface Area (TSA) = 6a²
• Volume = a³

3. Right Circular Cylinder (Radius = r, Height = h)

• Curved Surface Area (CSA) = 2πrh
• Total Surface Area (TSA) = 2πr(r + h)
• Volume = πr²h

4. Right Circular Cone (Radius = r, Vertical Height = h, Slant Height = l)

• Slant Height: l = √(r² + h²)
• Curved Surface Area (CSA) = πrl
• Total Surface Area (TSA) = πr(l + r)
• Volume = (1/3)πr²h

5. Sphere (Radius = r)

• Surface Area = 4πr²
• Volume = (4/3)πr³

6. Hemisphere (Radius = r)

• Curved Surface Area (CSA) = 2πr²
• Total Surface Area (TSA) = 3πr²
• Volume = (2/3)πr³

4. Solved Examples (Board-Exam Style)

Example 1: Two Cubes Joined Together

Question: 2 cubes, each of volume 64 cm³, are joined end to end. Find the surface area of the resulting cuboid.

Solution:

1. Let the edge of each cube be 'a'.

• Volume of cube = a³ = 64 cm³
• a = ³√64 = 4 cm

1. When two cubes of edge 4 cm are joined end to end, a cuboid is formed:

• Length (l) = 4 + 4 = 8 cm
• Breadth (b) = 4 cm
• Height (h) = 4 cm

1. Surface area of the resulting cuboid = 2(lb + bh + hl)

• = 2(8 × 4 + 4 × 4 + 8 × 4)
• = 2(32 + 16 + 32)
• = 2(80) = 160 cm²

Answer: 160 cm²

Example 2: Toy Formed by Cone and Hemisphere

Question: A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of the same radius. The total height of the toy is 15.5 cm. Find the total surface area of the toy.

Solution:

1. Radius of cone and hemisphere (r) = 3.5 cm = 7/2 cm.
2. Height of the hemisphere = radius = 3.5 cm.
3. Height of the cone (h) = Total height − Radius of hemisphere

• h = 15.5 − 3.5 = 12 cm

1. Slant height of the cone (l):

• l = √(r² + h²) = √((3.5)² + 12²)
• l = √(12.25 + 144) = √156.25 = 12.5 cm

1. Total surface area of the toy = CSA of cone + CSA of hemisphere

• TSA = πrl + 2πr²
• TSA = πr(l + 2r)
• TSA = (22/7) × (7/2) × [12.5 + 2(3.5)]
• TSA = 11 × [12.5 + 7]
• TSA = 11 × 19.5 = 214.5 cm²

Answer: 214.5 cm²

Example 3: Medicine Capsule

Question: A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends. The length of the entire capsule is 14 mm and the diameter is 5 mm. Find its surface area.

Solution:

1. Diameter = 5 mm, so radius (r) = 5/2 = 2.5 mm.
2. Length of cylindrical part (h) = Total length − (2 × radius)

• h = 14 − (2.5 + 2.5) = 14 − 5 = 9 mm

1. Total surface area of capsule = CSA of cylinder + 2 × (CSA of hemisphere)

• Surface Area = 2πrh + 2 × (2πr²)
• = 2πrh + 4πr² = 2πr(h + 2r)
• = 2 × (22/7) × (5/2) × [9 + 2(2.5)]
• = (110/7) × [9 + 5]
• = (110/7) × 14
• = 110 × 2 = 220 mm²

Answer: 220 mm²

Example 4: Volume of a Solid Cone on a Hemisphere

Question: A solid is in the shape of a cone standing on a hemisphere with both their radii being equal to 1 cm, and the height of the cone is equal to its radius. Find the volume of the solid in terms of π.

Solution:

1. Radius (r) = 1 cm, Height of cone (h) = 1 cm.
2. Volume of the solid = Volume of cone + Volume of hemisphere

• Volume = (1/3)πr²h + (2/3)πr³
• = (1/3)π × (1)² × (1) + (2/3)π × (1)³
• = (1/3)π + (2/3)π
• = (3/3)π = π cm³

Answer: π cm³

5. Common Mistakes and Quick Revision Points

• Mistake 1: Adding Total Surface Areas blindly.

Wrong: TSA of toy = TSA of cone + TSA of hemisphere.

Correct: TSA of toy = CSA of cone + CSA of hemisphere. The circular bases touch inside and are not part of the surface!

• Mistake 2: Mixing Slant Height (l) with Vertical Height (h).

In the cone formula for CSA (πrl), you must use the slant height l = √(r² + h²). In the volume formula (1/3)πr²h, you must use the vertical height h.

• Mistake 3: Confusing Diameter and Radius.

Questions often give the diameter. Always divide by 2 first to get the radius before plugging it into formulas.

• Mistake 4: Unit mismatch.

Ensure all measurements are in the same unit (e.g., all in cm or all in m) before starting calculations. If converting volume: 1 m³ = 1000 litres, 1 litre = 1000 cm³.

• Quick Revision Tip: Always draw a neat sketch of the combined solid and label r, h, and l clearly before writing the formula.