Editing 2509: Useful Geometry Formulas
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The radius of the circle, from the center to the right edge where it meets the ellipse, is labeled 'r'. In a textbook diagram of a sphere, the radius might be instead labeled with a diagonal line from the center to a different point on the ellipse, implying the generality that all points on that cross-section, and indeed on the whole spherical surface, are at the same radius from the center. However, this line would be shorter on the page than the actual radius, making it useless for the formula of the area of the 2D outer shape. | The radius of the circle, from the center to the right edge where it meets the ellipse, is labeled 'r'. In a textbook diagram of a sphere, the radius might be instead labeled with a diagonal line from the center to a different point on the ellipse, implying the generality that all points on that cross-section, and indeed on the whole spherical surface, are at the same radius from the center. However, this line would be shorter on the page than the actual radius, making it useless for the formula of the area of the 2D outer shape. | ||
− | + | Assuming an orthographic projection of a sphere, the area of the 2D shape on the page is the area of the circle, which is A = πr<sup>2</sup>. This is captioned below the figure. Real spheres are seen in a perspective projection, rather than an orthographic, and their visual area is then that of a slight ellipse rather than precisely a circle. | |
Coincidentally the area of the horizontal cross-section of the 3D sphere, as depicted by the ellipse, is also πr<sup>2</sup>, and a reader familiar with such diagrams might initially assume that this is what was meant. However, this does not extend to the other figures. | Coincidentally the area of the horizontal cross-section of the 3D sphere, as depicted by the ellipse, is also πr<sup>2</sup>, and a reader familiar with such diagrams might initially assume that this is what was meant. However, this does not extend to the other figures. |