Right Circular Cone

In the earlier grades, having studied the surface areas of basic shapes like cube, cuboid and cylinder, let's move on to study the surface area of cone.

We have upto now, been generating solids by stacking up congruent figures. A prisms is one such example. Let's look at another kind of solid which is not a prism. These kinds of solids are called pyramids.

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Activity: Cut out a right-angled triangle from a piece of paper. Paste the cutout on a long thick string along one of the perpendicular sides of the triangle i.e. the side that corresponds as the height. Now, hold the string with hands on either sides of the cutout and rotate the triangle using the string a number of times.

The shape that is generated is a right circular cone. A right angled triangle, when rotated to a full 360°180° degrees gives a right circular cone.

The'h' and 'r' denote the height and radius of the cone while 'l' is called the slant height of the cone i.e. the length of the curved surface's slope.

Since, the cone is right-angled, we also get that:

l2 = r2 + h2

We know that the base of the cone is a .

Thus, the area of the base can be evaluated. Now, how can the area of the entire curved surface be found?

Activity: Take a cone which doesn't have any operlapping surface. If unavailable, make a cone from a piece of paper. Cut out the cone along the slant-height .Upon opening it out and flattening the material, we will see a circular shape with a sector cut out of it (resembling a piece of cake which has a portion of it cutout).

Now, mark and cutout this circular shape into smaller triangles, all having their height cut out along the original slant height of the cone. Which in this case,is the radius of the circular shape. Now, individually, measure and calculate the area of each triangle.

Area of each triangle = 12 × base of each triangle × slant height

Thus we can say,

Area of the entire curved surface = sum of all the smaller triangles

= 12 x slant height x (b₁ + b₂ + .....)

= 12 x slant height x (circumference of the partial circle)

= 12 x slant height x (2πr)

= πr x slant height = πrl

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In a figure(cone), ∆AOB is a right angled triangle.

From ∆AOB

AB2 = AO2 + OB2

AB2 = h2 + r2

l2 = h2 + r2

l = h2+r2

Example 1 : Find the curved surface area of a right circular cone whose slant height is 10 cm and base radius is 7 cm.

Instructions

Now if the base of the cone is to be closed, then a circular piece of paper of radius r is also required whose area is πr2. Thus,

Volume of a cone is of the volume of the cylinder

Volume of a cone = 13πr2h