Cone

Definition:

Cone is the solid that is generated by rotating a line segment which is passing through a fixed point and making a constant angle with a fixed line.

In the above figure here , VO is a fixed line and VA is a rotating line which is making constant angle with VO.The point A would describe a circle with center O such that the line segment VO is perpendicular to the base.

VO is the height "h" and OA is the base radius "r"  VA is the slant height "l".

It is very clear VAO is right angled triangle and the right angle is at at O.

Since VAO is the right angled triangle by Pythagorean theorem

we have l² = h² + r²

from l² = h² + r² , we can get the value of one of the measurements if we know the value of the other two measurements. For example, if we have the height "h", and the radius "r" , we can easily determine the value of the slant height "l".

Example:

If the height and radius of a right circular conoid are 4cm and 3cm respectively,find the slant height.

Solution:

Let us plug the known values in to the equation.

l² = h² + r²

l² = 4² + 3²

l² = 16 + 9

l² = 25

so l = 5

that is slant height = 5 cm

Role of radius,slant height and height in finding area and volume:

Measurement of radius,height and slant height plays a vital role in finding curved surface area, total surface area and volumes.

Example

A heap of paddy is in the form of a conoid whose diameter is 4.2 m and height is 2.8 m. If the heap is to be covered exactly by a canvas to protect it from rain,then find the area of the canvas needed.

Solution:

Diameter of heap of paddy = 4.2 m

                     r = 4.2/2

                       = 2.1 m

height of paddy (h) = 2.8 m

             L² = r²  + h²

                L = √(2.1)² + (2.8)²

                L = √4.41 + 7.84

                L = √12.25

             L = √ 3.5 x 3.5

             L = 3.5 cm

Curved surface area of heap of paddy = Π r l

                                         = (22/7) x (2.1) x (3.5)

                                         = 22 x (2.1) x (0.5)

                                         = 23.1 cm²

Curved surface area of paddy = 23.1 cm²
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