Theory:

Right circular cone:
A right circular cone is a cone whose apex (top vertex of the cone) is perpendicular to the centre of the base of the circle.
cone_3 (1).svg
 
Properties of right circular cone:
1. A cone has only one apex.
 
2. A cone has no edges.
 
3. The base of the cone is circular.
 
4. The length of the sides of the cone from the apex to the circular base is the slant of the cone. It is denoted by the letter \('l'\).
 
5. The length of the line joining from the apex to the centre of the base is the height of the cone. It is denoted by the letter \('h'\).
Curved surface area of a cone:
cone_2 (1).svg
 
Let \(r\) be the radius and \(l\) be the arc length.
 
C. S. A. \(=\) \(\frac{\text{Arc length of the sector}}{\text{Circumference of the circle}} \times \text{Area of the circle}\)
 
\(=\) 2πl2πr×πr2
 
\(=\) \(\pi r l\)
 
Curved surface area of a cone \(=\) \(\pi r l\) sq. units
 
where \(r\) \(=\) radius of the base of the cone 
 
\(l\) \(=\) slant height of the cone
Slant height:
cone_3 (1).svg
 
\(AOB\) is a right-angled triangle, right-angled at \(O\).
 
Using Pythagoras theorem:
 
\(\text{Hypotenuse}^2 = \text{Base}^2 + \text{Height}^2\)
 
\(AB^2 = BO^2 + AO^2\)
 
\(l^2 = r^2 + h^2\)
 
\(l = \sqrt{r^2 + h^2}\) units
Total surface area of a cone:
T. S. A. \(=\) Curved surface area \(+\) Area of the base
 
\(=\) \(\pi r l\) \(+\) \(\pi r^2\)
 
\(=\) \(\pi r(l + r)\)
 
Total surface area of a cone \(=\) \(\pi r(l + r)\) sq. units
Oblique cone:
images.png
 
This is also a cone but not a right circular cone. Because the line joining the apex is not perpendicular to the centre of the base. It is called an oblique cone.