Cylinder

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.

First Figure

Second Figure

Third Figure

Fourth Figure

Fifth Figure

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 cm²

                                               = 64 cm²

(ii) The volume of the cylinder = (Area of the cross Section) × length

                                            = 64 cm² × 20 cm

                                            = 1280 cm³

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 cm²

(ii) Total surface area = lateral surface area + 2 × (Area of the Cross Section)

               = 288 cm² + 2 × (Area of the regular hexagon of side 4 cm)

               = 288 cm² + 2 × \(\frac{3√3}{2}\) × 4^2 cm²

               = (288 + 48√3) cm²

               = (288 + 83.04) cm²

                = 371.04 cm²

9th Grade Math

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