Question

Calculate the surface area of the cone

A) \[301.44{\text{ c}}{{\text{m}}^2}\]
B) \[301.44{\text{ cm}}\]
C) \[301.44{\text{ m}}\]
D) \[301.44{\text{ }}{{\text{m}}^2}\]

Answer 1

Hint: As we can see the only difference in the option is the unit. But before answering that let us first derive the solution by using the total surface area formula of the cone. The total surface area of the cone comprises the curved surface area and a circle.

Complete step-by-step solution:
Given,
Diameter of the circle
\[d = 12{\text{ cm}}\]
So, the radius will be half of the diameter which is,
\[r = \dfrac{d}{2} = \dfrac{{12}}{2} = 6{\text{ cm}}\]
Slant height, \[l = 10{\text{ cm}}\]
The total surface area of the cone comprises an area of curved surface area and the area of a circle
Curved surface area \[ = \pi rl\]
Area of circle \[ = \pi {r^2}\]
Combining both, the formula for the total surface area of the cone is
\[TS = \pi rl + \pi {r^2}\]
The value of \[\pi \] that we will take is 3.14
Substituting the values of r and l in the equation, we get
\[TS = [3.14 \times 6{\text{ cm}} \times 10{\text{ cm]}} + [3.14 \times {(6{\text{ cm)}}^2}]\]
\[TS = [3.14 \times 60{\text{ c}}{{\text{m}}^2}{\text{]}} + [3.14 \times 36{\text{ c}}{{\text{m}}^2}]\]
Multiplying the value of \[\pi \] in the equation, we get
\[TS = 188.4{\text{ c}}{{\text{m}}^2} + 113.04{\text{ c}}{{\text{m}}^2}\]
\[TS = 301.44{\text{ c}}{{\text{m}}^2}\]
It will be \[{\text{c}}{{\text{m}}^2}\] because our values are in cm and surface area is always measured in square units.
Hence, the surface area of the cone is \[301.44{\text{ c}}{{\text{m}}^2}\].
Therefore, the correct option is A.

Note: Remember that we will always find the total surface area when asked to find the surface area. The term "curved" will always be used in the question when referring to the curved surface area. We simply included one circle because the cone only has one. If the measurements were given in meters, the solution would have been in meter squares.