Class 10 Maths · Chapter 12 · Topic 1 of 5
First identify the solid and the measurement
Choose whether you need exposed surface or occupied volume.
Ideas you’ll use
- Surface area and volume
- Surface area has square units. Volume has cubic units. Capacity is the usable internal volume of a container.
Understand it, one idea at a time
Choose an idea. Follow its reason, calculation and clue in the picture.
A closed cylinder has a curve and two discs
- Blue
- Cylinder outline; ellipse represents a circular end seen obliquely.
- Orange / violet
- Cone or hemispherical constituent where shown.
- Green dashed
- Joining disc or cavity opening; not an exposed flat face.
Name the solid and dimensions
For a cylinder use radius r and perpendicular height h, with one length unit.
r=3 cm; h=8 cmIn the picture: Identify the curved wall and flat discs.
Read all the reasoning together
Name the solid and dimensions
For a cylinder use radius r and perpendicular height h, with one length unit.
r=3 cm; h=8 cmUnroll the curved surface
Its rectangle has width 2πr and height h.
curved area=2πrh=48π cm²Check which ends are included
A closed cylinder includes two discs; an open-top one includes only its bottom.
closed area=66π cm²; open-top area=57π cm²Measure the inside volume
Multiply base area by height, rather than adding faces.
volume=πr²h=72π cm³
Choose whether you need exposed surface or occupied volume.
Worked example: An original cylinder
r=3 cm, h=8 cm ⇒ curved area 48π cm²; closed total area 66π cm²; volume 72π cm³An open-top cylinder has area 2πrh+πr² if its bottom is included. The shape alone does not tell you which surface a question asks to paint.
Read the supporting explanation
For cuboid length L, breadth b, height h: area 2(Lb+bh+hL), volume Lbh. For cube edge a: area 6a², volume a³. For a right circular cylinder: curved area 2πrh, total area 2πrh+2πr², volume πr²h.
Go deeper: Cone, sphere and hemisphere toolkit
Right cone: slant l=√(r²+h²), curved area πrl, total area πrl+πr², volume πr²h/3. Sphere: area 4πr², volume 4πr³/3. Hemisphere: curved area 2πr², total area 3πr², volume 2πr³/3. The hemisphere’s flat disc is included only in its total area.
Your turn
Try the idea yourself. Show your reasoning, then compare.
Question
Should the answer to “how much water does it hold?” have cm² or cm³?
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