Before getting into the concept of the frustum of a cone, give a quick glance at the below important points, which makes you understand the concept much better.
CONE: A cone is a 3 – D geometrical figure which has a circular base (flat base) that narrows smoothly from its base to a point called the apex.
The volume of a cone = 1/3 πr²h cm³
The surface area of the cone = πrl+ πr2
Conversion Of Solid From One Shape To Another
We all have seen candles. They are in a cylindrical shape. But you may also see candles in different shapes like a square, sphere, and in many other different shapes. To have different shapes of the candle, just heat it and collect the wax from it and place it in the container, which is of the shape you want. In a similar way, the objects can be converted into other shapes with different sizes.
Let’s consider an example.
A clay cone has a height of 24 cm and a base radius of 6cm. It is then reshaped into a sphere. Now, what will be the radius of the reshaped sphere?
Therefore, the radius of the sphere is 6 cm.
From the above discussion, we have understood the concept of conversion of solid to different shapes. So we use this format to explain the frustum of a cone.
Frustum Of A Cone
Now, take a right circular cone. Now remove a small portion, i.e., cut the given cone by a plane parallel to its base. Now, a small cone is formed on one side of that plane. The part that is now leftover on the other side of the plane is called a frustum of the cone.
A frustum is a Latin word meaning "piece cut off. "
Formulae Of Frustum Of A Cone
|1. The volume of the frustum of a cone||=||1/3πh (r1² + r2² + r1r2)|
|2. The Curved surface area of a frustum of a cone||=||(r1 + r2) l|
|3. The total surface area of the frustum of a cone||=||l (r1 + r2) + r1² + r2²|
Note: l =√h²+(√r1 - √r2)²
A metal bucket has a frustum shape and is attached to a hollow cylindrical base. The diameter of the 2 circular ends is 25cm and 45cm, respectively. The height of the bucket is 40cm. The base of the cylinder is 6cm. Find the area of the bucket and the amount of water it can hold.
The total height = 40 cm (the base height is also incorporated in it)
Therefore the frustum height of the cone = (40 – 6) = 34 cm.