Cylinder, Cone and Sphere (Surface Area and Volume)

347. The Hollow Cylinder's Secret · Volume and surface area formulas

The cross-section area always equals π(R² - r²).

hhCross-section (material) area = π(R² − r²) = π(22500 − 10000) = 39270Cross-section (material) area = π(R² − r²) = π(22500 − 10000) = 39270R (outer) = 150R (outer) = 150r (inner) = 100r (inner) = 100height h = 200height h = 200Rrh
A hollow cylinder (tube) has outer radius R and inner radius r. Its cross-section is a ring of area π(R² − r²), so its volume = π(R² − r²)h — the material between the two surfaces.

Stuck? Ask Guru

Selina ICSE: Cylinder, Cone and Sphere (Surface Area and Volume)

What this lesson covers

Try to break it

Drag R for the outer radius and r for the inner radius. The material cross-section area is the annulus π(R² − r²) = π(R − r)(R + r). Push r toward R and the wall thins out to zero (no material). Push r toward 0 and the cylinder becomes solid. The material area depends on the difference of squares, not the difference of radii.

How you build it

Build a hollow tube: draw the outer and inner rims at both ends as flat ellipses, then drop the outer and inner walls of height h.

  • Ellipse tool: drag a WIDE, FLAT ellipse near the top — the outer rim of radius R.
  • Ellipse tool: draw the bottom outer rim directly below the first — it keeps the same size R. The gap between them is the height h.
  • Ellipse tool: draw a SMALLER flat ellipse inside the top outer rim, same centre — the inner opening of radius r.
  • Ellipse tool: draw the bottom inner rim directly below the top inner rim — it keeps the same size r, inside the bottom outer rim.
  • Segment tool: join the LEFT edge of the top outer rim to the left edge of the bottom outer rim — the outer wall, height h.
  • Segment tool: join the RIGHT edges of the two outer rims — the other outer wall.
  • Segment tool: join the LEFT edges of the two inner rims — the wall of the hole.
  • Segment tool: join the RIGHT edges of the two inner rims. The four rims and four walls now form the hollow tube — cross-section π(R² − r²).

The proof, step by step

Prove that the cross-section area of a hollow cylinder is π(R² − r²).

  • Cross-section is an annulus. Area = πR² - πr² = π(R² - r²).
  • External curved surface area = 2πRh.
  • Internal curved surface area = 2πrh.
  • Total Surface Area = External + Internal + 2 × Cross-section = 2πRh + 2πrh + 2π(R² - r²).
  • Volume of material = External volume - Internal volume = πR²h - πr²h = π(R² - r²)h.

Worked example

A hollow cylindrical pipe has an external radius of 5 cm and an internal radius of 3 cm. If the length of the pipe is 14 cm, find the volume of the material used in making the pipe.

V = π(R² - r²)h = (22/7)(5² - 3²) × 14 = (22/7) × 16 × 14 = 704 cm³.

  • 704 cm³ — correct
  • 720 cm³
  • 680 cm³
  • 750 cm³
Hold to talk

Subscription Status