346. The Cylinder's Anatomy · Surface, space, and slices
The cylinder's dimensions and properties update consistently as you drag R and H.
What this lesson covers
Try to break it
Drag R to grow or shrink the radius and H to lengthen the cylinder. Volume = πr²h, total surface = 2πr(r + h), lateral surface = 2πrh — all three readouts adjust together. Take any horizontal slice and you get a circle of radius r, identical to the top and bottom: that's the uniform cross-section signature of a cylinder.
How you build it
Construct a cylinder from its two circular bases joined by two straight sides of height h.
- Ellipse tool: drag a WIDE, FLAT ellipse near the bottom — the base of the cylinder (a circle of radius r seen at an angle).
- Ellipse tool: draw the top ellipse directly above the base — it keeps the same size. The gap between the two ellipses is the height h.
- Segment tool: join the LEFT edge of the base to the left edge of the top — the left side of the cylinder, of height h.
- Segment tool: join the RIGHT edge of the base to the right edge of the top. The two bases and two sides now form a cylinder of radius r and height h — curved surface 2πrh, total surface 2πr(r + h), volume πr²h.
The proof, step by step
Prove that the radius and height fully describe a cylinder.
- Total Surface Area = Lateral Surface Area + 2 × Area of Base Circle.
- Lateral Surface Area = 2πrh.
- Volume = πr²h.
- The cross-section of a cylinder cut perpendicular to its height is a circle congruent to the base.
Worked example
A cylindrical tank has a radius of 7 cm and height of 15 cm. Find its lateral surface area. (Use π = 22/7)
LSA = 2πrh = 2 × (22/7) × 7 × 15 = 660 cm². The lateral surface area excludes the top and bottom circles, covering only the curved side.
- 330 cm²
- 660 cm² — correct
- 1038.5 cm²
- 462 cm²