# 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
Triangle3 sides$$A=\frac12 h l$$
(Same for all)
Proof 2
Equal-sided...Equal sides
Isosceles...3Two equal sides
Right-angled...Two perpendicular sides
Obtuse-angled...2An angle larger than $90^\circ$
Acute-angled...1No angles larger than $90^\circ$
3D
PyramidEdged base and ends in a peak
(regardless of no. of edges)
$$V=\frac13 A_{base}h$$Proof 7
2D
SquareEqual sides and angles$$A=s^2$$Proof 1
RectangleEqual angles$$A=lw$$Proof 1
RhombusParallel and equal sides$$A=\frac12 d_1d_2$$Proof 9
ParallelogramParallel sides$$A=lw$$Proof 10
TrapeziumTwo parallel sides$$A=\frac12 (w_1+w_2)l$$Proof 10
3D
CubeA dice$$V=s^3$$Proof 1
BoxEqual angles$$V=lwh$$Proof 1
PrismEdged 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
CircleA ring$$L=2\pi r$$Proof 3
... sectionA piece of a ring$$L=\theta r$$Proof 3 7
2D
Circle / disc6A pizza$$A=\pi r^2$$Proof 4
... sectorA pizza slice$$A=\frac12 \theta r^2$$Proof 4
EllipsisA ‘squeezed’ pizza$$A=\pi r_1r_2$$Proof 5
3D
CylinderA column$$V=\pi r^2 h$$Proof 6
ConeCircular base and
ends in a peak
$$V=\frac13 \pi r^2 h$$Proof 7
Ball / sphereA globe$$V=\frac43 \pi r^3$$Proof 11
EllipsoidA ‘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:

1. List of Geometric Shapes’ (web page), Crispin Pennington, Math Salamanders, www.math-salamanders.com/list-of-geometric-shapes.html (accessed May 10th, 2019)
2. 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)
3. Online Etymology Dictionary’ (dictionary), Douglas Harper, www.etymonline.com