How do you calculate the surface area of a cone?

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How do you calculate the surface area of a cone?

Geometry
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|dw:1434127734377:dw| unravelling the cone will give us the surface area, so we do the following
|dw:1434127987313:dw| if we make a circle and take a ratio and find the areas, we will be left with the area of the figure drawn in this image with \[A = \pi r y\] so now to take the surface area of a cone, we will have \[\huge SA = \pi r y + \pi r ^2\]
We can arrive at the same formula with a bit of calculus. |dw:1434138596071:dw|

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Revolve the curve about the x-axis to generate a surface: |dw:1434138690167:dw| The surface area is given by the integral \[2\pi\int_0^h\frac{r}{h}x\,dx\] where \(\dfrac{r}{h}x\) is the radius of each circular cross-section taken at a particular \(0\le x\le h\), and so multiplying by \(2\pi\) and taking the integral along this interval gives the "lateral" surface area as the infinite sum of circumferences. \[A_{\text{lateral}}=\frac{2\pi r}{h}\int_0^h x\,dx=\frac{\pi r}{h}(h^2-0^2)=\pi rh\] Add the area of the "base", which is a circle with radius \(r\) to get the formula \[A_{\text{cone}}=\pi rh+\pi r^2\]
Ah yes, I was going to use calculus as well but not sure whether or not Preetha knows it, nice one Siths!

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