## Preetha one year ago How do you calculate the surface area of a cone?

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1. Astrophysics

|dw:1434127734377:dw| unravelling the cone will give us the surface area, so we do the following

2. Astrophysics

|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 $\huge SA = \pi r y + \pi r ^2$

3. anonymous

We can arrive at the same formula with a bit of calculus. |dw:1434138596071:dw|

4. anonymous

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 $A_{\text{cone}}=\pi rh+\pi r^2$

5. Astrophysics

Ah yes, I was going to use calculus as well but not sure whether or not Preetha knows it, nice one Siths!