Properties of Cylinder 

Some of the important properties of the cylinder are as follows:

• The bases of the cylinder are always congruent and parallel to each other.
• If the axis of the cylinder is a right angle to the base and the bases are exactly over each other, then it is called as “Right Cylinder”.
• If one of the bases of the cylinder is displayed sideways, and the axis does not produce the right angle to the bases, then it is called “Oblique Cylinder”.
• If the bases are circular, then it is called a right circular cylinder.
• The best alternative to the circular base of a cylinder is an ellipse. If the base of the cylinder is elliptical in shape, then it is called an “Elliptical Cylinder”.
• If the locus of a line moving parallel and fixed distance from the axis, a circular cylinder is produced.
• A cylinder is similar to the prism since it has the same cross-section everywhere.

Cylinder (Right Circular Cylinder)

Let the radius of the base and height of the right circular cylinder be ‘R’ and ‘H’ respectively.

• Volume = π R² H
• Curved Surface area = 2 π R H
• Total surface area = 2 π R H + 2 π R²

Hollow Cylinder (Hollow Right Circular Cylinder)

Let the inner radius of the base, outer radius of the base and height of the hollow right circular cylinder be ‘r’, ‘R’ and ‘H’ respectively.

• Volume = π H (R² – r²)
• Curved Surface area = 2 π R H + 2 π r H = 2 π H (R + r)
• Total surface area = 2 π H (R + r) + 2 π (R² – r²)

Cone

A cone is a shape formed by using a set of line segments or the lines which connects a common point, called the apex or vertex, to all the points of a circular base(which does not contain the apex). The distance from the vertex of the cone to the base is the height of the cone. The circular base has measured value of radius. And the length of the cone from apex to any point on the circumference of the base is the slant height. Based on these quantities, there are formulas derived for surface area and volume of the cone. In the figure you will see, the cone which is defined by its height, the radius of its base and slant height.

Let the radius of the base, slant height and height of the cone be ‘R’, ‘L’ and ‘H’ respectively.

• L² = R² + H²
• Volume = π R² H / 3
• Curved Surface area = π R L
• Total surface area = π R L + π R²

Examples

Question: Find the volume of the cone if radius, r = 4 cm and height, h = 7 cm.

Solution: By the formula of volume of the cone, we get,

V = πr²h/3

V = (1/3) × (22/7) × 4² × 7

V = 117.33 Cubic Cm

Question: What is the total surface area of the cone with the radius = 3 cm and height = 5 cm ?

Solution: By the formula of the surface area of the cone, we know,

Area = πr(l + r)

Since, slant height l = √(r²+h²) = √(3²+5²) = √(9+25) = √34

Therefore,

Area, A = π × 3(√34 + 3) = π × 3(5.83 + 3) = π × 3(8.83) = 83.22 Cm²