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:
[tex]\begin{gathered} A=\pi r^2 \\ A=3.14\times(13)^2 \\ A=530.66\text{ dm}^2 \end{gathered}[/tex]
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
[tex]\begin{gathered} sin\theta\degree=\frac{opp.}{hyp.} \\ sin45\degree=\frac{13}{l} \\ \frac{1}{\sqrt{2}}=\frac{13}{l} \\ l=13\sqrt{2} \\ l=13\times1.414 \\ l\approx18..4\text{ dm} \end{gathered}[/tex]
The total surface area of the cone is given by the formula:
[tex]A=\pi r(r+l)[/tex]
Where r= radius of the base
l = slant height
Thus the total surface area
[tex]\begin{gathered} A=3.14\times13(13+18.4) \\ A=40.82(31.4) \\ A=1281.75\text{ dm}^2 \end{gathered}[/tex]
Final Answer:
Area of base =
[tex]530.66\text{ dm}^2[/tex]
Side of surface l =
[tex]18.4\text{ dm}[/tex]
The total surface area of the cone
[tex]=1281.75\text{ dm}^2[/tex]
