Mathematical tables: typical geometry related formulas
Tables show typical formulas related to geometry such as surface area of various geometric plane shapes, disk curcumference, sphere volume etc.

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Disk and circle

 Name Formula Legend Disk area surface Show source$S:=\pi\cdot{ R}^{2}$ S - surface area of the disk,R - disk radius (length of line segments from its center to its perimeter),$\pi$ - pi number (math constant approximately equal to 3.14159). Disk or circle circumference Show source$L:=2~\pi\cdot R$ L - disk circumference (linear distance around circle or disk),R - disk radius (length of line segments from its center to its perimeter),$\pi$ - pi number (math constant approximately equal to 3.14159). Disk radius from area surface Show source$R:=\sqrt{\frac{ S}{\pi}}$ R - disk radius (length of line segments from its center to its perimeter),S - surface area of the disk,$\pi$ - pi number (math constant approximately equal to 3.14159). Disk radius from area Show source$L:=2~\sqrt{\pi\cdot S}$ L - disk circumference (linear distance around circle or disk),S - surface area of the disk,$\pi$ - pi number (math constant approximately equal to 3.14159). Disk (circle) circumference from circumference Show source$R:=\frac{ L}{2~\pi}$ R - disk radius (length of line segments from its center to its perimeter),L - disk circumference (linear distance around circle or disk),$\pi$ - pi number (math constant approximately equal to 3.14159). Disk (circle) area from circumference Show source$S:=\frac{{ L}^{2}}{4~\pi}$ S - surface area of the disk,L - disk circumference (linear distance around circle or disk),$\pi$ - pi number (math constant approximately equal to 3.14159).

Surface areas of plane shapes

 Name Formula Legend Disk area surface Show source$S:=\pi\cdot{ R}^{2}$ S - surface area of the disk,R - disk radius (length of line segments from its center to its perimeter),$\pi$ - pi number (math constant approximately equal to 3.14159). Parallelogram area Show source$S:= a\cdot h$ S - area of the parallelogram,a - length of the parallelogram base,h - height of the parallelogram. Rectangle area Show source$S:= a\cdot b$ S - area of the rectangle,a - length of the first rectangle side,b - length of the second rectangle side. Rectangle diagonal Show source$d:=\sqrt{{ a}^{2}+{ b}^{2}}$ d - length of the rectangle diagonal,a - length of the first rectangle side,b - length of the second rectangle side. Rectangle area from diagonal Show source$S:=\frac{1}{2}\cdot{ d}^{2}\cdot\sin\left( \alpha\right)$ S - area of the rectangle,d - length of the rectangle diagonal,$\alpha$ - smaller angle between rectangle diagonals. Rhombus area Show source$S:=\frac{1}{2}\cdot e\cdot f$ S - area of the rhombus,e - length of the first rhombus diagonal,f - length of the second rhombus diagonal. Square area Show source$S:={ a}^{2}$ S - area of the square,a - length of the square side. Square area from diagonal Show source$S:=\frac{{ d}^{2}}{2}$ S - area of the square,d - length of the square diagonal. Trapezoid area Show source$S:=\frac{\left( a+ b\right)}{2}\cdot h$ S - area of the trapezoid,a - length of the first trapezoid base,b - length of the second trapezoid base,h - height of the trapezoid. Triangle area (general) Show source$S:=\frac{1}{2}\cdot a\cdot h$ S - area of the triangle,a - length of the triangle base,h - height of the triangle. Equilateral triangle area Show source$S:=\frac{ a\cdot\sqrt{3}}{2}$ S - area of the equilateral triangle,a - length of the equilateral triangle side. Isosceles triangle area Show source$S:=\frac{ b}{4}\cdot\sqrt{4~{ a}^{2}-{ b}^{2}}$ S - area of the isosceles triangle,a - length of the isosceles triangle base,b - length of the isosceles triangle side.

Circumference of shapes

 Name Formula Legend Disk or circle circumference Show source$L:=2~\pi\cdot R$ L - disk circumference (linear distance around circle or disk),R - disk radius (length of line segments from its center to its perimeter),$\pi$ - pi number (math constant approximately equal to 3.14159). Parallelogram circumference Show source$L:=2\cdot\left( a+ b\right)$ L - circumference of the parallelogram,a - length of the first parallelogram side,b - length of the second parallelogram side. Rectangle circumference Show source$L:=2\cdot\left( a+ b\right)$ L - circumference of the rectangle,a - length of the first rectangle side,b - length of the second rectangle side. Rhombus circumference Show source$L:=4\cdot a$ L - circumference of the rhombus,a - length of the rhombus side. Square circumference Show source$L:=4\cdot a$ L - circumference of the square,a - length of the square side. Triangle circumference (general) Show source$L:= a+ b+ c$ L - circumference of the triangle,a - length of the triangle first side,b - length of the triangle second side,c - length of the triangle third side. Equilateral triangle circumference Show source$L:=3\cdot a$ L - circumference of the equilateral triangle,a - length of the equilateral triangle side. Isosceles triangle circumference Show source$L:= a+2\cdot b$ L - circumference of the isosceles triangle,a - length of the isosceles triangle base,b - length of the isosceles triangle side.

Parallelogram

 Name Formula Legend Parallelogram area Show source$S:= a\cdot h$ S - area of the parallelogram,a - length of the parallelogram base,h - height of the parallelogram. Parallelogram circumference Show source$L:=2\cdot\left( a+ b\right)$ L - circumference of the parallelogram,a - length of the first parallelogram side,b - length of the second parallelogram side.

Renctangle

 Name Formula Legend Rectangle area Show source$S:= a\cdot b$ S - area of the rectangle,a - length of the first rectangle side,b - length of the second rectangle side. Rectangle circumference Show source$L:=2\cdot\left( a+ b\right)$ L - circumference of the rectangle,a - length of the first rectangle side,b - length of the second rectangle side. Rectangle diagonal Show source$d:=\sqrt{{ a}^{2}+{ b}^{2}}$ d - length of the rectangle diagonal,a - length of the first rectangle side,b - length of the second rectangle side. Rectangle area from diagonal Show source$S:=\frac{1}{2}\cdot{ d}^{2}\cdot\sin\left( \alpha\right)$ S - area of the rectangle,d - length of the rectangle diagonal,$\alpha$ - smaller angle between rectangle diagonals.

Rhombus

 Name Formula Legend Rhombus area Show source$S:=\frac{1}{2}\cdot e\cdot f$ S - area of the rhombus,e - length of the first rhombus diagonal,f - length of the second rhombus diagonal. Rhombus circumference Show source$L:=4\cdot a$ L - circumference of the rhombus,a - length of the rhombus side.

Square

 Name Formula Legend Square area Show source$S:={ a}^{2}$ S - area of the square,a - length of the square side. Square circumference Show source$L:=4\cdot a$ L - circumference of the square,a - length of the square side. Square diagonal Show source$d:= a~\sqrt{2}$ d - length of the square diagonal,a - length of the square side. Square area from diagonal Show source$S:=\frac{{ d}^{2}}{2}$ S - area of the square,d - length of the square diagonal.

Trapezoid

 Name Formula Legend Trapezoid area Show source$S:=\frac{\left( a+ b\right)}{2}\cdot h$ S - area of the trapezoid,a - length of the first trapezoid base,b - length of the second trapezoid base,h - height of the trapezoid.

Triangle

 Name Formula Legend Triangle area (general) Show source$S:=\frac{1}{2}\cdot a\cdot h$ S - area of the triangle,a - length of the triangle base,h - height of the triangle. Triangle circumference (general) Show source$L:= a+ b+ c$ L - circumference of the triangle,a - length of the triangle first side,b - length of the triangle second side,c - length of the triangle third side. Equilateral triangle area Show source$S:=\frac{ a\cdot\sqrt{3}}{2}$ S - area of the equilateral triangle,a - length of the equilateral triangle side. Equilateral triangle circumference Show source$L:=3\cdot a$ L - circumference of the equilateral triangle,a - length of the equilateral triangle side. Isosceles triangle area Show source$S:=\frac{ b}{4}\cdot\sqrt{4~{ a}^{2}-{ b}^{2}}$ S - area of the isosceles triangle,a - length of the isosceles triangle base,b - length of the isosceles triangle side. Isosceles triangle circumference Show source$L:= a+2\cdot b$ L - circumference of the isosceles triangle,a - length of the isosceles triangle base,b - length of the isosceles triangle side.

Equilateral triangle

 Name Formula Legend Equilateral triangle area Show source$S:=\frac{ a\cdot\sqrt{3}}{2}$ S - area of the equilateral triangle,a - length of the equilateral triangle side. Equilateral triangle circumference Show source$L:=3\cdot a$ L - circumference of the equilateral triangle,a - length of the equilateral triangle side.

Isosceles triangle

 Name Formula Legend Isosceles triangle area Show source$S:=\frac{ b}{4}\cdot\sqrt{4~{ a}^{2}-{ b}^{2}}$ S - area of the isosceles triangle,a - length of the isosceles triangle base,b - length of the isosceles triangle side. Isosceles triangle circumference Show source$L:= a+2\cdot b$ L - circumference of the isosceles triangle,a - length of the isosceles triangle base,b - length of the isosceles triangle side.

Some facts

• Geometry is one of two oldest fields of mathematics (next to arithmetic).
• Geometry examines geometric shapes and their relationships.
• Due to the kind of shapes we deal with, we divide the geometry into two parts:
• planimetry - the part dealing with flat shapes, i.e. those that can be drawn in a 2D plane like square, circle, etc.,
• stereometry - the part dealing with spatial solids, i.e. three-dimensional shapes such as cube or cylinder.
• The origins of geometry go back to ancient times. The father of geometry in the form we know today is Greek philosopher Euclid. About 300 BC he prioritized knowledge about geometry, which resulted in a dissertation "Elements".
• Euclid's "Elements" are considered one of the first theoretical works in mathematics. Euclid, ordering the previous knowledge, indicated a few the most basic laws (so-called axioms), and then he used them to derive all existing geometry as today's mathematicians do. This is why his work is considered a breakthrough not only for the development of geometry, but mathematics in general.
• The axioms adopted by Euclid were as follows:
• 1. To draw a straight line from any point to any point.
• 2. To produce a finite straight line continuously in a straight line.
• 3. To describe a circle with any centre and distance.
• 4. That all right angles are equal to one another.
• 5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.
• Nowadays, geometry based on the above postulates is called Euclidean geometry. However, over time, mathematicians began to study geometries based on other axioms, removing or modifying selected points from the original list used by Euclid. Such geometries are, for example, Riemann's geometry (removing the Euclidian postulate 5.), used to formulate General relativity of Einstein or so-called pointless geometry.