Ninth Grade Mathematics: Surface Area & Volume of Right Circular Cylinders
The cylinder is the natural three-dimensional extension of the circle, representing the solid generated by revolving a rectangle around one of its sides. In this chapter, you will explore the geometry of right circular cylinders, derive formulas for curved and total surface area by unrolling the curved surface, and master volume and hollow cylinder calculations.
The Geometry of a Right Circular Cylinder
A cylinder is defined as a solid bounded by a cylindrical curved surface and two parallel planar circular bases. A cylinder is called a right circular cylinder if the line segment joining the centers of the two circular bases is perpendicular to the planes of the bases.
Anatomy of a Right Circular Cylinder:
. - ~ ~ - .
. ' r ' . Top Circular Base (Area = pi * r^2)
*---------------------*
| | |
| | |
| | Height |
| | (h) |
| | |
*---------------------*
. ' ' . Bottom Circular Base (Area = pi * r^2)
' - ~ ~ - .
The radius of the circular bases is the base radius r, and the perpendicular separation between the bases is the height h (also called the length or altitude).
Unrolling the Curved Surface: The Derivation of CSA
To understand the curved surface area (CSA) of a cylinder, imagine slicing the curved metal shell vertically and flattening it onto a table. The curved surface unrolls into a perfect rectangle:
UNROLLING THE CYLINDRICAL SURFACE
. - ~ . +-----------------------+
. ' r ' . | |
*-----------* | | Height (h)
| | ---(Slice & Unroll)---> | |
*-----------* +-----------------------+
. ' _ _ ' . <-- Circumference 2*pi*r -->
The vertical height of the unrolled rectangle is equal to the cylinder's height h. The horizontal width of the unrolled rectangle is equal to the circumference of the circular base: 2 * pi * r.
Because the area of any rectangle is width * height:
Curved Surface Area (CSA):
CSA = Circumference of Base * Height = (2 * pi * r) * h
CSA = 2 * pi * r * h
To find the Total Surface Area (TSA), add the areas of the two circular bases (top and bottom) to the curved surface area:
Total Surface Area (TSA):
TSA = Curved Surface Area + Area of Two Circular Bases
= 2 * pi * r * h + 2 * (pi * r^2)
TSA = 2 * pi * r * (h + r)
Volume of a Right Circular Cylinder
By Cavalieri's Principle, the volume of any uniform prism or cylinder is the product of its base area and perpendicular height:
Volume of Cylinder = Area of Circular Base * Height
Volume = pi * r^2 * h
Hollow Cylinders (Cylindrical Pipes)
A hollow cylinder (such as a metal pipe) has an outer radius R, an inner radius r, and length h:
Hollow Cylinder (Pipe)
. - ~ ~ - .
. ' R r ' .
/ . - ~ . \
| | * | |
| | | | Height (h)
| | | |
\ . _ _ . /
. ' ' .
' - ~ ~ - .
Hollow Cylinder Formulas:
Internal Curved Surface Area: CSA_inner = 2 * pi * r * h
External Curved Surface Area: CSA_outer = 2 * pi * R * h
Area of each Circular Ring: Area_ring = pi * (R^2 - r^2)
Total Surface Area (TSA): TSA = 2*pi*R*h + 2*pi*r*h + 2*pi*(R^2 - r^2)
Volume of Material in Pipe: Volume = pi * (R^2 - r^2) * h
Chapter Exercises
Exercise 1: A closed cylindrical water tank has base radius 7 meters and height 10 meters. Calculate its Curved Surface Area, its Total Surface Area, and its internal water-holding capacity in liters (1 m^3 = 1,000 liters; use 22/7 for pi).
Exercise 2: The curved surface area of a right circular cylinder is 880 square cm, and its height is 20 cm. Find the radius of its base and its volume.
Exercise 3: A metallic cylindrical pipe is 21 cm long. Its outer diameter is 10 cm and its inner diameter is 6 cm. Find the volume of metal used to manufacture the pipe, and calculate the total surface area of the pipe.
Solutions and Explanations
Solution 1: Identify dimensions: r = 7 m, h = 10 m. Curved Surface Area is CSA = 2 * pi * r * h = 2 * (22/7) * 7 * 10 = 2 * 22 * 10 = 440 square meters. Total Surface Area is TSA = 2 * pi * r * (h + r) = 2 * (22/7) * 7 * (10 + 7) = 44 * 17 = 748 square meters. The volume is V = pi * r^2 * h = (22/7) * (7^2) * 10 = (22/7) * 49 * 10 = 22 * 7 * 10 = 1,540 cubic meters. Convert to liters: Capacity = 1,540 * 1,000 = 1,540,000 liters.
Solution 2: We are given CSA = 880 cm^2 and h = 20 cm. The formula for CSA is 2 * pi * r * h: 880 = 2 * (22/7) * r * 20. Simplify: 880 = (880 / 7) * r. Multiply by 7/880: r = 7 cm. The base radius is 7 cm. Now calculate the volume: V = pi * r^2 * h = (22/7) * (7^2) * 20 = (22/7) * 49 * 20 = 22 * 7 * 20 = 3,080 cubic cm.
Solution 3: For the pipe: length h = 21 cm. Outer diameter = 10 cm, so outer radius R = 10 / 2 = 5 cm. Inner diameter = 6 cm, so inner radius r = 6 / 2 = 3 cm. Volume of metal is V = pi * (R^2 - r^2) * h = (22/7) * (5^2 - 3^2) * 21 = (22/7) * (25 - 9) * 21 = 22 * 16 * 3 = 1,056 cubic cm. To find Total Surface Area, sum external CSA, internal CSA, and the two circular base rings: External CSA = 2 * pi * R * h = 2 * (22/7) * 5 * 21 = 2 * 22 * 5 * 3 = 660 cm^2; Internal CSA = 2 * pi * r * h = 2 * (22/7) * 3 * 21 = 2 * 22 * 3 * 3 = 396 cm^2; Area of two rings = 2 * pi * (R^2 - r^2) = 2 * (22/7) * 16 = 704 / 7 approx 100.57 cm^2. Total Surface Area = 660 + 396 + 100.57 = 1,156.57 square cm.