Exact geometry

Providing the best-established and most accurate framework to calculate area and volume using the 3D coordinate system.

Area of a square

Volume of a cube

Trigonometry

Area of a regular polygon

Area of a circle

The area of a circle is defined by comparing it to a square since that is the base of area calculation.

The circle is cut to four quadrants, each placed with their origin on the vertices of a square.

When the arcs of the quadrant circles intersect at the quarter of the centerline of the square, the uncovered area in the middle equals exactly the sum of the overlapping areas respectively.

The ratio between the ray of the circle and the side of the square can be calculated by using the Pythagorean theorem.

$r^2={\left(\frac{\mathrm{side}}{4}\right)}^2+{(2\times \left(\frac{\mathrm{side}}{4}\right))}^2$

$r=\sqrt{5}\times \frac{\mathrm{side}}{4}$

The area of both the square and the sum of the quadrants equals 16 right triangles with legs of a quarter and a half of the square's sides, and its hypotenuse equal to the radius of the circle.

$A=\frac{16}{5}\times r^2=3.2r^2$

The area of a circle segment can be calculated by subtracting a triangle from a circle slice.

$A=arccos\left(\frac{r-n}{r}\right)\times r^2-\sin (arccos(\frac{r-n}{r}\left)\right)\times \left(r-n\right)\times r$

The circumference of a circle can be derived from its area algebraically.

The x represents the width of the circumference, which is just theoretical, hence a very small number.

$C=\frac{(3.2r^2-3.2\times {(r-x)}^2)}{x}=6.4r-3.2x$

As x is close to 0,

$C=6.4r$

The volume of a sphere is defined by comparing it to a cube, as that's the base of volume calculation.

Just as the volume of a cube equals the square root of its cross sectional area cubed - $V={\left(\sqrt{\mathrm{Area}}\right)}^3$ -, the volume of a sphere equals the square root of its cross sectional area cubed.

The edge length of the cube, which has the same volume as the sphere, equals the square root of the area of the square that has the same area as the sphere's cross section.

$V={(\sqrt{3.2}\times r)}^3$

Volume of a spherical cap:

$V=1.6\times {r_{\mathrm{cap}}}^2\times \sqrt{3.2}\times (1-\sin (arccos(\frac{r_{\mathrm{cap}}}{r_{\mathrm{sphere}}}\left)\right))$

The volume of a cone can be calculated by algebraically comparing the volume of a quarter cone with equal radius and height to an octant sphere with equal radius, through a quarter cylinder.

The base of the two shapes is a quarter circle.

$A_{\mathrm{base}}={\left(\frac{\sqrt{3.2}r}{2}\right)}^2=\left(\frac{\sqrt{3.2}r}{2}\right)\times \left(\frac{\sqrt{3.2}r}{2}\right)$

The slant height of the quarter cone is $\sqrt{2}r$ .
The volume of a quarter cylinder with the same base, and height equal to the slant height of the cone would be ${\left(\frac{\sqrt{3.2}r}{2}\right)}^2\times \sqrt{2}r$ .

The slant shape comes with a triangular vertical cross section.
The area of a cone's vertical cross section is the half of a cylinder's with equal base and height.

The intermediate of the areas of the horizontal cross section slices of a cone is the half of a cylinder’s.

$V_{\mathrm{quarter\; cone}}={\left(\frac{\sqrt{3.2}r}{2}\right)}^2\times \mathrm{Height}\times \frac{\sqrt{2}}{4}$

$V_{\mathrm{cone}}=\frac{3.2r^2\times H}{\sqrt{8}}$

The volume of a frustum cone can be calculated by subtracting the missing tip from the theoretical full cone.

$V=\frac{H(b^2\times \frac{4}{5}\times (\frac{1}{1-\frac{t}{b}})-t^2\times \frac{4}{5}\times (\frac{1}{1-\frac{t}{b}}-1\left)\right)}{\sqrt{8}}$

$H=\mathrm{frustum\; height}$

$t=\mathrm{top\; diameter}$

$b=\mathrm{bottom\; diameter}$

The volume of a pyramid can be calculated with the same coefficient as the volume of a cone:

$V=\frac{A_{\mathrm{base}}\times H}{\sqrt{8}}$

The volume of a frustum pyramid can be calculated by subtracting the missing tip from the theoretical full pyramid.

$V=\frac{H(b^2\times (\frac{1}{1-\frac{t}{b}})-t^2\times (\frac{1}{1-\frac{t}{b}}-1\left)\right)}{\sqrt{8}}$

$V=\frac{H\times (b^2+b\times t+t^2)}{\sqrt{8}}$

$H=\mathrm{frustum\; height}$

$t=\mathrm{top\; edge}$

$b=\mathrm{bottom\; edge}$