Revolving a point about a line will get us a circle.
Solid: A composite solid made by subtracting a cone from a cylinder. The radius of the cylinder is 3 units and the height is 4 units. Surface Area: 48Ď€ square units
Practice makes perfect
We will start by analyzing the given triangle.
Let's revolve the given triangle about the line y=3. Remember that revolving a point about the line will get us a circle.
We see that the revolved triangle creates a composite solid made by subtracting a cone from a cylinder. The radius of the cylinder is 3 units and the height is 4 units. We are asked to find the surface area of the revolved solid. First, let's find the slant height of the cone, l.
The surface area of the composite solid consists of three parts.
The lateral area of the cone, L.A._(cone).
The lateral area of the cylinder, \text{L.A.}_\text{cyl}.
The base area of the cylinder, B.
Let's find them!
Radius
r= 3
Height
h= 4
Slant Height
l= 5
\text{L.A.}_\text{cone}
Ď€ r l=Ď€ ( 3)( 5)=15Ď€
\text{L.A.}_\text{col}
2Ď€ r h=2Ď€ ( 3)( 4)=24Ď€
B
Ď€ r^2=Ď€ ( 3)^2=9Ď€
Finally, we will find the surface area of the composite solid, S.A..
\begin{gathered}
\textcolor{darkviolet}{\text{S.A.}}=\text{L.A.}_\text{cone}+\text{L.A.}_\text{col}+B \\
\Downarrow \\
\textcolor{darkviolet}{\text{S.A.}}=\textcolor{darkorange}{15\pi}+\textcolor{darkorange}{24\pi}+\textcolor{darkorange}{9\pi}=\textcolor{darkviolet}{48\pi}
\end{gathered}
The surface area of the revolved solid is 48Ď€ square units.
