How to find out the surface area of the right circular cone

Surface area of the cone = πr(l+r) 

Where r is the radius of the circular top of the cone BD as shown in the fig  FIG-1. And l is the lateral height of the cone AD as shown in FIG-1.

What is a cone

A cone is an object as shown in the FIG-1. A cone should have a point “A” as shown in the above fig. And this point is called the vertex of the cone. The cone should also have the height. From the fig. the height is “AB”. In the top there should be a circular shape as shown in the fig. The diameter of the circle is “CD”. The diameter and the height should be in perpendicular. Then only it can call as a right circular cone. the top shape also should be in perfect circle in order to call it “circular cone”. Now we can also observe from the fig., the curved surface of the cone is formed with a slant height. The lateral height in this case is AC and AD.

Curved surface area of a cone

As shown in the above fig. we can make the curved surface area of a cone. The curved surface area of the con in the fig-2a can be made of with circular paper in the fig-2b. Her the missing part of the fig-2a is because of the missing part if the fig-2b.

We can cut the circular paper in number of pieces shown in the fig-2b in such a way that we get a smaltrangle with base b1. If the number of the triangle is in hundreds, then the base b1,b2,b3,…… of the triangle can be considered straight line. And the height of the triangle is the lateral height of the cone. that is nothing but the radius OB  of the circle. And if we arrange all that small triangles we can get the curved surface area of the cone.

Now we know the area of each small triangle =   ½*b1*l

Area of all small triangle = ½*b1*l + ½*b1*l + ½*b1*l + ……….

½l(b1+b2+b3+…..)

Now if we add all the b1+b2+b3 ….. we will get the circumference of the circle that is nothing but 2πr.

so the curved surface area =½l*2πr 

=πrl

Now we have to consider the area of the top circle of the con . Which is πr².

total surface area of the cone = πrl+πr²  = πr (l+r)

so the total surface area of the right angle circular cone is  πr(l+r) 

Where r is the radius of the cone and l is the lateral height of the cone.

IMPORTANT notes

From the above fig-3, we can see the triangle ABC  in the cone. Here AB is the height of the cone, BD is the radius of the cone and AD is the slant height. So if we have only the radius and height of the cone we can find out the slant height. Because slant height is required to find out the surface are. By using pythagoras theorem we can find out the slant height of the cone by the below equation

AD² = AB²+BD²

AD = √(AB²+BD²)

Where AD  is the slant height of the cone, AB  is the height of the cone and BD is the radius of the cone.

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