Artificial intelligent assistant

How do I find the surface area of an angled conic base? Thank you for viewing my question. I need help creating a formula for finding the surface area of a conic base. (eg. I install a flood light on my roof, I want to know how much surface area it will cover). What I know: 1. 25ft - The height of the light source above the ground 2. 30 degrees - The angle at which the light source is pointed towards the earth 3. 25 degrees - The vertical light beam width 4. 25 degrees The horizontal light beam width Vertical light beam width is the angle in which the light emits horizontally from the lens of the light source. This creates a left and right limit. Horizontal light beam width is the angle in which the light emits vertically from the lens of the light source. This creates a top and bottom limit. example of vertical light beam width and horizontal light beamwidth

OK, let $\theta$ be the angle the light is tilted, $\alpha$ be the horizontal beam divergence (width), $\beta$ be the vertical beam divergence, and $h$ be the height above the ground.

You should be able to convince yourself with right triangles that the horizontal beam width on the ground is

$$h \left[\tan{\left(\theta+\frac{\alpha}{2} \right)} - \tan{\left(\theta-\frac{\alpha}{2} \right)}\right] = \frac{2 h \tan{\frac{\alpha}{2}}}{\cos^2{\theta}-\sin^2{\theta} \tan^2{\frac{\alpha}{2}}} $$

There is an analogous expression for $\beta$. The area of the beam (assuming a rectangular shape, add whatever geometrical factors you feel exist) is then

$$A = \frac{4 h^2 \tan{\frac{\alpha}{2}} \tan{\frac{\beta}{2}}}{\left(\cos^2{\theta}-\sin^2{\theta} \tan^2{\frac{\alpha}{2}}\right)\left(\cos^2{\theta}-\sin^2{\theta} \tan^2{\frac{\beta}{2}}\right)} $$

Using the given numbers, I get $A \approx 1123\, \text{ft}^2$.

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