Lessons 23–24 · 2 lessons · M. A. Mirzaahmedov, Sh. N. Ismailov, A. Q. Amanov (algebra and analysis), B. Q. Haydarov (geometry). Mathematics Grade 11, Parts 1 and 2, 1st edition. ZAMIN NASHR, Tashkent, 2018
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The surface and volume of a cylinder
Textbook, Part 1: pp. 172–183
GoalLearn the elements and sections of a cylinder and the formulas for its lateral surface, total surface and volume, and use them in problems.
A right circular cylinder is produced by rotating a rectangle about one of its sides (the axis): the side parallel to the axis is a generator (its length l equals the height h) and the other two sides sweep out two equal bases (discs) of radius r. A plane parallel to the axis cuts the cylinder in a rectangle; the section through the axis is the axial section, with sides 2r and h. The net is a rectangle with sides 2πr and h together with two discs, so S_lat = 2πr · h and S_total = 2πr · h + 2πr² = 2πr(h + r). The volume is V = S_base · h = πr²h; the formula is obtained by passing to the limit between the volumes of inscribed and circumscribed prisms. The base of an inscribed prism is a polygon inscribed in the base circle; that of a circumscribed prism is circumscribed about it. When a body is immersed in a liquid, the volume of the raised water equals the volume of the body: V = πr² · Δh.
Worked examples
r = 4, h = 9: S_lat = 2π · 4 · 9 = 72π, S_total = 2π · 4 · (9 + 4) = 104π, V = π · 16 · 9 = 144π. If the axial section of a cylinder has diagonal 17 and the base diameter is 8, the height is h = √(289 − 64) = 15 and r = 4: V = π · 16 · 15 = 240π, S_lat = 2π · 4 · 15 = 120π.
A stone is dropped into a cylindrical vessel of radius 10 cm and the water level rises by 2 cm. The stone’s volume is V = π · 10² · 2 = 200π ≈ 628 cm³. A section parallel to the axis at distance d from it: for r = 5, h = 6, d = 3 the chord is 2√(25 − 9) = 8 and the section area is 8 · 6 = 48.
Class activity
“Water level”: take a clear cylindrical glass (with adult supervision) and a small stone or metal part. Measure the inner radius of the glass and the rise of the water level, and compute the part’s volume with V = πr²Δh. Compare with the volume computed from its dimensions.
Practice
1
Find the volume of a cylinder with r = 6 and h = 10.
360π
2
A cylinder has a base circumference of 14π cm and height 5 cm. Find its lateral surface.
70π cm²
3
A cylinder has generator 9 and the perimeter of its axial section is 30. Find the base radius.
3
4
Why is the total surface of a cylinder 2πr(h + r)?
The lateral surface unfolds into a rectangle with sides 2πr (the base circumference) and h, with area 2πrh. The two base discs have area πr² each, 2πr² in total. The sum is 2πrh + 2πr² = 2πr(h + r).