Question

Find the surface area of the conical hat. If its slant height is three times the radius, the base diameter of a hat is 4 inches. (Use\[\pi = 3\])

Answer 1

Hint: Area is the measure of the size of a two-dimensional flat-surface whereas surface area
refers to the size on a surface represented on a three-dimensional surface. In the case of the conical hat,
we will have to find both the base area as well as the curved surface area, which is also known as the
lateral surface area. The lateral surface of a cone is made when a sector is rotated along with its altitude
hence, the lateral surface area is equal to the area of a sector\[LSA = \pi rl\], where \[l\] is the radius of
the sector or the slant height of the conical hat and \[r\]is the arc length or radius of the conical hat. In a
cone, the base is in the shape of a circle. Hence we use the formula of area of the circle to find the area
of the base, $A = \pi {r^2}$.

Complete step by step answer:
The surface area of conical hat = Area of the base + Lateral surface area
\[SA = \pi {r^2} + \pi rl\]
Given the diameter of the base of the hat \[d = 4inches\], hence the radius \[r = \dfrac{d}{2} = 2{\text{
}}inches\]and slant height of the hat is given three times the radius \[l = 3r = 3 \times 2 = 6{\text{
}}inches\]
Thus,
Area of base of the hat \[A = \pi {r^2} = 3 \times {\left( 2 \right)^2} = 3 \times 4 = 12i{n^2}\]
Lateral surface area \[LSA = \pi rl = 3 \times 2 \times 6 = 36i{n^2}\]
Therefore the Surface area of the cone\[ = \pi {r^2} + \pi rl = 12 + 36 = 48i{n^2}\].

Note: We calculate the area of a body when the body is represented on a two-dimensional surface, whereas in the case of three-dimensional objects, we measure their Surface area. Curved surface area is the area of
a three-dimensional surface which can be seen or can be touched without deforming the shape of the
object.