A solid with uniform cross section perpendicular to its length (or height) is a cylinder. The cross section may be a circle, a triangle, a square, a rectangle or a polygon. A can, a pencil, a book, a glass prism, etc., are examples of cylinders. Each one of the figures shown below is a cylinder.
The second figure shown above is a cuboid, which is also a cylinder. The cylinder depicted in the last figure is also called prism.
The formule for the volume and the surface area of a cylinder are the same as those of solids with uniform cross section.
Volume of a Cylinder = (Area of the cross section) × length (or height or breadth)
= A × h
Lateral surface area of a Cylinder = (Perimeter of the cross section) × length (or height or breadth)
= P × h
Total surface area of a Cylinder = Lateral surface area + Sum of the areas of the two plane ends
= P × h + 2 × A
Solved Problems on Volume and Surface Area of Cylinder:
1. The cross section of a cylinder is a trapezium whose parallel sides measure 10 cm and 6 cm, and the distance between the parallel sides is 8 cm. If the cylinder is 20 cm long, find (i) the area of the cross section; and (ii) the volume of the cylinder.
Solution:
(i) The area of the cross section = area of the trapezium
= \(\frac{1}{2}\) (10 + 6)8 cm^{2}
= 64 cm^{2}
(ii) The volume of the cylinder = (Area of the cross Section) × length
= 64 cm^{2} × 20 cm
= 1280 cm^{3}
2. The cross section of a cylinder is a regular hexagon of side 4 cm and its height measures 12 cm. Find its (i) lateral surface area, and (ii) total surface area.
Solution:
(i) The perimeter of the cross section
= Perimeter of a regulator hexagon of side 4 cm
= 6 × 4 cm
= 24 cm.
Therefore, lateral surface area of the cylinder
= (Perimeter of the cross section) × height
= 24 cm × 12 cm
= 288 cm^{2}
(ii) Total surface area = lateral surface area + 2 × (Area of the Cross Section)
= 288 cm^{2} + 2 × (Area of the regular hexagon of side 4 cm)
= 288 cm^{2} + 2 × \(\frac{3√3}{2}\) × 4^2 cm^{2}
= (288 + 48√3) cm^{2}
= (288 + 83.04) cm^{2}
= 371.04 cm^{2}
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