Editing 2509: Useful Geometry Formulas
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This illustration is commonly used to depict a 3D cylinder or right circular prism. In this case, the upper ellipse represents the "visible" part of the top circular surface, with its "depth" shorter than its "width" due to foreshortening, and the lower part of the lower ellipse represents the "front" edge of the lower surface; the dotted half of the lower ellipse represents the occluded "back" edge of the lower surface. | This illustration is commonly used to depict a 3D cylinder or right circular prism. In this case, the upper ellipse represents the "visible" part of the top circular surface, with its "depth" shorter than its "width" due to foreshortening, and the lower part of the lower ellipse represents the "front" edge of the lower surface; the dotted half of the lower ellipse represents the occluded "back" edge of the lower surface. | ||
− | To add to the confusion, the upper ellipse has its major axis labeled 'd' which usually denotes the diameter of a circular surface, while the lower ellipse has its | + | To add to the confusion, the upper ellipse has its major axis labeled 'd' which usually denotes the diameter of a circular surface, while the lower ellipse has its semimajor axis labeled 'r' which similarly denotes a radius, even though the ellipses drawn have neither diameter nor radius. The 'h' denoting height is also used for both rectangles and solid objects. While 'd' in this case is required for the area calculation of the 2D shape, in textbooks only 'r' may be marked and the arrow may be offset at a diagonal rather than in line with any figurative axis, to imply its applicability to any angle of radius. |
The non-overlapping parts of the 2D shape are composed of the rectangle formed by the major axes of the two ellipses and the vertical lines, plus half of the top ellipse and half of the bottom ellipse. The area of the rectangle is dh, and the area of an ellipse with semimajor axis d/2 and semiminor axis r is πrd/2. The total area is A = d(πr/2 + h), which is captioned below the figure. | The non-overlapping parts of the 2D shape are composed of the rectangle formed by the major axes of the two ellipses and the vertical lines, plus half of the top ellipse and half of the bottom ellipse. The area of the rectangle is dh, and the area of an ellipse with semimajor axis d/2 and semiminor axis r is πrd/2. The total area is A = d(πr/2 + h), which is captioned below the figure. |