Resource: Geometric shapes
Overview of many usual geometric shapes.^{[1,2]} Formulas for areas, volumes and a few lengths/circumferences are shown for the shapes. Symbols used:
- $L$ is path length, $A$ area, $V$ volume,
- $A_{base}$ base/bottom area,
- $l$ length, $w$ width, $h$ height,
- $s$ side length (when all sides are equal),
- $w_1$ and $w_2$ the two different trapezium side lengths,
- $d_1$ and $d_2$ the two diagonals (line between opposite corners),
- $r$ radius, $r_1$, $r_2$ and $r_3$ half-axes in ellipses/ellipsoids, and
- $\theta$ angle turned.
3-edged (triangular) shapes | ||||
---|---|---|---|---|
2D | ||||
Triangle | 3 sides | $$A=\frac12 h l$$ (Same for all) | Proof 2 | |
Equal-sided... | Equal sides | |||
Isosceles...^{3} | Two equal sides | |||
Right-angled... | Two perpendicular sides | |||
Obtuse-angled...^{2} | An angle larger than $90^\circ$ | |||
Acute-angled...^{1} | No angles larger than $90^\circ$ | |||
3D | ||||
Pyramid | Edged base and ends in a peak (regardless of no. of edges) | $$V=\frac13 A_{base}h$$ | Proof 7 |
4-edged (quadrilateral)^{4} shapes | ||||
---|---|---|---|---|
2D | ||||
Square | Equal sides and angles | $$A=s^2$$ | Proof 1 | |
Rectangle | Equal angles | $$A=lw$$ | Proof 1 | |
Rhombus | Parallel and equal sides | $$A=\frac12 d_1d_2$$ | Proof 9 | |
Parallelogram | Parallel sides | $$A=lw$$ | Proof 10 | |
Trapezium | Two parallel sides | $$A=\frac12 (w_1+w_2)l$$ | Proof 10 | |
3D | ||||
Cube | A dice | $$V=s^3$$ | Proof 1 | |
Box | Equal angles | $$V=lwh$$ | Proof 1 | |
Prism | Edged column (regardless of no. of edges) | $$V=A_{base}h$$ | Proof 6 |
These above were 3- and 4-sided shapes and we could continue with 5-sided, 6-sided etc. They all belong to the group called polygons.^{5}
Round (circular) shapes | ||||
---|---|---|---|---|
1D | ||||
Circle | A ring | $$L=2\pi r$$ | Proof 3 | |
... section | A piece of a ring | $$L=\theta r$$ | Proof 3 ^{7} | |
2D | ||||
Circle / disc^{6} | A pizza | $$A=\pi r^2$$ | Proof 4 | |
... sector | A pizza slice | $$A=\frac12 \theta r^2$$ | Proof 4 | |
Ellipsis | A ‘squeezed’ pizza | $$A=\pi r_1r_2$$ | Proof 5 | |
3D | ||||
Cylinder | A column | $$V=\pi r^2 h$$ | Proof 6 | |
Cone | Circular base and ends in a peak | $$V=\frac13 \pi r^2 h$$ | Proof 7 | |
Ball / sphere | A globe | $$V=\frac43 \pi r^3$$ | Proof 11 | |
Ellipsoid | A ‘squeezed’ ball | $$V=\frac43 \pi r_1r_2r_3$$ | Proof 11 |
Many more formulas for parameters in these shapes exist, such as relationships between the angles and the sides in a triangle (triangle-specific geometry is called trigonometry (see the Trigonometry skill in the Forces discipline).
References:
- ‘List of Geometric Shapes’ (web page), Crispin Pennington, Math Salamanders, www.math-salamanders.com/list-of-geometric-shapes.html (accessed May 10th, 2019)
- ‘Geometric Shapes: List, Definition, Types of Geometric Shapes’ (web page), Toppr, 2018, www.toppr.com/guides/maths/basic-geometrical-ideas/basic-geometrical-shapes (accessed May 10th, 2019)
- ‘Online Etymology Dictionary’ (dictionary), Douglas Harper, www.etymonline.com
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