Editing 2509: Useful Geometry Formulas

{{comic
| number    = 2509
| date      = August 30, 2021
| title     = Useful Geometry Formulas
| image     = useful_geometry_formulas.png
| titletext = Geometry textbooks always try to trick you by adding decorative stripes and dotted lines.
}}

==Explanation==
This comic showcases area formulas for the areas of four two-dimensional geometric shapes which each have extra dotted and/or solid lines making them look like illustrations for 3-dimensional objects. The first, a simple equation for the area of a circle, the second an equation for the area of a triangle with a semi-elliptic base, the third an equation for the area of a rectangle with an elliptical base and top, and the fourth an equation for the area of a hexagon consisting of two opposing right-angled corners and two parallel diagonal lines connecting their sides. In each case, only the area formed by the outline of each shape is calculated.

Similar illustrations are commonly found in geometry textbooks, which are used to depict three-dimensional figures on a two-dimensional page. They commonly make use of slanted lines to indicate edges receding into the distance and dashed lines to indicate an edge occluded by nearer parts of the solid. The joke is that the formulae given here are for the area of each two-dimensional shape within its outer solid lines, not for the surface area or volume of the illustrated 3D object (as would be shown in the geometry textbook). The title text continues the joke by claiming that the dotted lines are simply decorative.

The illustrations depict the following plane or solid figures, depending on the interpretation.

; Top Left - Circle with an inscribed ellipse, or Sphere
This illustration is commonly used to depict a three-dimensional sphere, with the ellipse representing a "horizontal" or axial cross-section through the center; the solid lower half of the ellipse represents the "front" of the circumference of this cross-section, while the dotted upper half represents the "back" of the same section, which would be occluded from view if this were a solid 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.  

The 3D sphere commonly depicted by this drawing would have a volume of <sup>4</sup>/<sub>3</sub>&nbsp;πr<sup>3</sup> and a surface area of 4πr<sup>2</sup>.  

; Top Right - Ellipse with symmetrical diagonal lines, or Cone
This illustration is commonly used to depict a three-dimensional right circular cone, with the lower half of the ellipse representing the "front edge" of the bottom surface, and the upper half representing the occluded "back edge".  However such drawings would usually not use both 'a' and 'b' to describe the radius of the base of the cone, which is drawn as an ellipse due to foreshortening.  Alternatively, the drawing could depict a right elliptical cone.

Randall approximates the area of the 2D shape on the page as the sum of the area of the triangle formed by the major axis of the ellipse and the two lines, and half of the area of the ellipse (<sup>π</sup>/<sub>2</sub>&nbsp;ab) since most of the upper half of the ellipse overlaps the triangle.  The equation for this area is A = 1/2 πab + bh.  This is captioned below the figure.

The actual area of a picture of a cone is not Randall's approximation, because the sides connect at the points on the ellipse where they can spread widest and form tangents to the ellipse, and such points are a little higher than those which define the major axis. This is most obvious in cases when h is only a little larger than a. The area can be computed to be exactly A = b (a arccos(-a/h)) + √(h<sup>2</sup>-a<sup>2</sup>)).

The 3D right circular cone commonly depicted by this drawing would have a volume of πr<sup>2</sup>h/3 where r=a=b.  The area of the "lower" surface would be πr<sup>2</sup>, while the surface area of the upper conical surface would be πr√(h<sup>2</sup> + r<sup>2</sup>).  Neither of these areas can correspond with the caption in the comic, nor does the total surface area (the sum of these two).

If we do not assume that a = b, this drawing could also depict a right elliptic cone.  The volume of the elliptic cone would be <sup>π</sup>/<sub>3</sub>&nbsp;abh.  The area of the lower surface would be πab and the area of the curved upper surface would be <br>2a√(b<sup>2</sup>&nbsp;+&nbsp;h<sup>2</sup>)&nbsp;<sub>0</sub>∫<sup>1</sup>&nbsp;√(<sup>a²h²(t²-1)&nbsp;-&nbsp;b²(a²+h²t²)</sup>/<sub>a²(t²-1)(b²+h²)</sub>)&nbsp;dt. 

; Bottom Left - Two ellipses joined vertically, or Cylinder
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 semiminor 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.

A 3D right circular prism (cylinder) would have a volume of πr<sup>2</sup>h and a surface area of 2πr<sup>2</sup> + πdh, or 2πr(r + h) since in this case d = 2r.  The area of each flat surface would be πr<sup>2</sup>.  If we do not assume d = 2r, then the lateral surface area of the right elliptic cylinder is 4h&nbsp;<sub>0</sub>∫<sup>1</sup>&nbsp;√(<sup>1&nbsp;-&nbsp;t²(1-4r²/d²)</sup>/<sub>1&nbsp;-&nbsp;t²</sub>)&nbsp;dt. The volume is <sup>π</sup>/<sub>2</sub>&nbsp;rdh. 

; Bottom Right - Parallel Hexagon, or Prism
This illustration is commonly used to depict a rectangular prism, with 'b' denoting the 'breadth', 'd' the 'depth' and 'h' the 'height'.  However, the labeled angle θ, which is necessary for the area calculation of the 2D shape, would not normally be used in a diagram of a rectangular prism, as all angles are assumed to be right angles.  A rhomboidal prism could be accurately described by this diagram with the assumption that the 'base' parallelogram is perpendicular to the 'front' and that the only non-right angle is θ.  In that case 'd' would not accurately describe the depth of the solid, which would be d sin θ.

The area of the 2D shape comprises the rectangle at the lower left, the parallelogram above it, and the parallelogram on the right.  The area of the rectangle representing the front face of the prism is bh. The area of the upper parallelogram is db&nbsp;sin&nbsp;θ. The area of the right parallelogram is dh&nbsp;cos&nbsp;θ.  The equation for this area is  A = bh + d(b sinθ + h cosθ) as is given below the figure. 

The surface area of the prism would be 2bh&nbsp;+&nbsp;2db sin θ&nbsp;+&nbsp;2dh. The volume is bdh sin θ. Assuming a 3D shape, θ can be artificially altered by the projection; the assumption could be made that θ is 90 degrees, and sin θ is 1 (and therefore can be eliminated from the formulas), but since θ is marked, such an assumption might not be valid.

In the history of the development of computer-generated 3D graphics, calculations of the apparent visual area taken up by the projection of a volume may have been useful in occlusion-like optimizations, where each drawn pixel may be passed through many fragment shaders.

==Transcript==
:[Four figures in two rows of two, each being a common two-dimensional representation of a three-dimensional object, with solid lines in front and dotted lines behind. Each figure has some labeled dimensions represented with arrows and a formula underneath indicating its area. Above the four figures is a header:]
:Useful geometry formulas

:[Top left; A circle with an inscribed concentric ellipse sharing its horizontal diameter. The edge of the ellipse above the major axis is drawn with a dotted line, while the lower edge is drawn with a solid line, similar to textbook depictions of a 3D sphere. The shared radius/semi-major axis to the right of the center is drawn as an arrow and labeled 'r'.  ]
:A = πr²

:[Top right; An ellipse with horizontal major axis, plus two straight lines: one from each end of the major axis, up to a point vertical to the center of the ellipse, so that the major axis of the ellipse (not drawn) and the two lines would form an isosceles triangle with a vertical axis of symmetry. The upper edge of the ellipse above the major axis is drawn with a dotted line, while the lower edge is drawn with a solid line, similar to textbook depictions of a right elliptical cone, or more commonly a right circular cone. The semi-minor axis of the ellipse is drawn with an arrow down from the center and labeled 'a' and the semi-major axis is similarly drawn to the right of the center and labeled 'b'.  To the right of the shape, the height of the isosceles triangle is drawn using arrows, and labeled 'h'.]
:A = 1/2 πab + bh

:[Bottom left; Two ellipses of the same dimensions, with major axes horizontal, drawn vertically one above the other, with vertical lines connecting each end of the major axis of the top ellipse to the corresponding points on the bottom ellipse.  The upper edge of the bottom ellipse above the major axis is drawn with a dotted line, while the lower edge is drawn with a solid line, similar to textbook depictions of a right elliptical prism or, more commonly, a right cylinder (circular prism). Inside the shape, the major axis of the upper ellipse is drawn as a double-ended arrow and labeled 'd'.  The semi-minor axis of the lower ellipse is drawn as an arrow down from the center and labeled 'r'. To the right of the shape, the length of the vertical lines is replicated using arrows and labeled 'h'. ]
:A = d(πr/2 + h)

:[Bottom right; Two rectangles of the same vertical and horizontal dimensions, drawn with one offset diagonally to the upper right of the other, with diagonal lines connecting the corresponding vertices, forming a hexagon with opposite sides parallel.  The upper right rectangle has its left and bottom sides drawn with dotted lines, and a similar dotted line is used connecting the bottom left corner of the two rectangles, similar to textbook depictions of rhomboid-based right prisms, or more commonly rectangular prisms.  Outside the shape, the bottom edge of the lower rectangle is redrawn below the shape with arrows and labeled 'b'. The length of the left edge is similarly redrawn to the left and labeled 'h'. The length of the diagonal line connecting the upper left corners of the two rectangles is similarly redrawn on the top left using arrows and labeled 'd'. The acute angle between the bottom edge of the lower rectangle, and the dotted diagonal connecting the two lower left corners, is labeled 'θ']
:A = bh + d(b sinθ + h cosθ)

{{comic discussion}}

[[Category:Math]]