- #1

- 766

- 49

$$A = 2πr^2 \cdot (1 – cos (\frac{θ}{2} )$$

However, I have another way but I don’t understand why this isn’t correct.

The circular area can be considered as a bulging base of a cone. The top of the cone emerging from the center of the sphere. Like this:

If we cut this cone in the middle, we’d get this in a 2D plane:

Here, ##r## doesn't have to be equal to ##R##, and ##θ## is in radians. If I divide the angle ##θ## of the cone by 2 and then multiply it by ##R##, I’d get the radius ##r## of the circle cap on the sphere, thus: ##\frac{θR}{2} = r##. And with that radius ##r## of that circle cap, I should be able to calculate its area. What I don’t get now is why the area of that circular cap is then not equal to:

$$π \cdot (\frac{θR}{2})^2$$

Is there a way to explain and prove why a circular cap on a sphere doesn’t have an area equal to ##πr^2##, even though it's a circle?