Geometry Tutorial Part 8

Community

Campus Placements

Author: Sureshbala

CONE

A cone is equivalent to a right pyramid whose base is a circle. The lateral surface of a cone does not consist of triangles like in a right pyramid but is a single curved surface.

If r is the radius of the base of the cone, h is height of the cone and l is the slant height of the cone, then we have the relationship (Fig. 1.41)

$Volume=\cfrac {1}{3} \pi r^2 h$
$\text {Curved Surface Area }=\pi r 1$
$\text {Total Surface Area }=\pi r 1+ \pi r^2 =\pi r(1+r)$

A cone can be formed by taking the sector of a circle and joining together its straight edges. If the radius of the sector is R and the angle of the sector is $\theta$ , then we have the following relationships between the length of the arc and area of the sector on the one hand and base perimeter of the cone and curved surface area of the cone on the other hand.

Radius of the sector = Slant height of the cone i.e., R=1

Length of the arc of the sector = Circumference of the base of the cone

$\cfrac {\theta}{360^{\circ}}*2*\pi r \Rightarrow r= \cfrac {\theta}{360^{\circ}}*R$

and Area of the sector = Curved surface area

(Actually, from this last equation, substituting the values from the first two equations, we can get the curved surface area of the cone, which is what is given previously as equal to $\pi r 1$ )

CONE FRUSTUM

If a cone is cut into two parts by a plane parallel to the base, the portion that contains the base is called the frustum of a cone.

If r is the top radius; R, the radius of the base ; h the height and l the slant height of a frustum of a cone (Fig. 1.42),then

$\text {Lateral Surface Area of the cone} =\pi 1(R+r)$
$\text {Total Surface Area } =\pi(R^2+r^2+R.1+R.1)$
$Volume=\cfrac {1}{3}\pi h(R^2+Rr+r^2)$
1^2=(R-r)^2+h^2

If H is the height of the complete cone from which the frustum I cut, then from similar triangles, we can write the following relationship.

$\cfrac {r}{R}=\cfrac {H-h}{H}$

A bucket that is normally used in a house is a good example of the frustum of a cone. The bucket is actually the inverted from the frustum that is shown in the figure above.

FRUSTUM OF A PYRAMID

A pyramid left after cutting of a portion at the top by a plane parallel to the base is called a frustum of a pyramid.

If A_1 is the area of base; A_2 the area of the top and h, the height of the frustum

$\text {Volume of frustum } = \cfrac{1}{3}*h*(A_1+A_2+\sqrt{A_1A_2})$
$\text {Lateral Surface Area}=\cfrac {1}{2}* \text {Sum of preimeters of base and top *Slant height}$ $\text {Total Surface Area =Lateral Surface Area}+A_1+A_2$

SPHERE

Any point on the surface of a sphere is equidistant from the centre of the sphere. This distance is the radius of the sphere