Question

The picture shows a cone.Calculate the areas of the base of the cone, the side surface and the whole surface.

Answer 1

Given:

The radius of the circle = 13 dm

And an angle = 45 degree

Required:

To Calculate the areas of the base of the cone, the side surface, and the whole surface.

Explanation:

The base of the cone is a circle and the area of the circle is given by the formula:

`\begin{gathered} A=\pi r^2 \\ A=3.14*(13)^2 \\ A=530.66\text{ dm}^2 \end{gathered}`

The given triangle is the right triangle so to find the side surface use the following trigonometric identity:

Let the side of the surface = l dm

`\begin{gathered} sin\theta\degree=(opp.)/(hyp.) \\ sin45\degree=(13)/(l) \\ (1)/(√(2))=(13)/(l) \\ l=13√(2) \\ l=13*1.414 \\ l\approx18..4\text{ dm} \end{gathered}`

The total surface area of the cone is given by the formula:

`A=\pi r(r+l)`

Where r= radius of the base

l = slant height

Thus the total surface area

`\begin{gathered} A=3.14*13(13+18.4) \\ A=40.82(31.4) \\ A=1281.75\text{ dm}^2 \end{gathered}`

Final Answer:

Area of base =

`530.66\text{ dm}^2`

Side of surface l =

`18.4\text{ dm}`

The total surface area of the cone

`=1281.75\text{ dm}^2`