For in GOD we live, and move, and have our being. - Acts 17:28

The Joy of a Teacher is the Success of his Students. - Samuel Dominic Chukwuemeka

# Mensuration Formulas

I greet you this day,
These are the formulas I used in developing the mensuration calculators. Some of them may not be exactly what you see in your textbooks. However, some of them are the same formulas.
Most likely, you will not see some of the formulas here in any textbook. This is because I derived those formulas.
If you want to see the derivations of some of the formulas, please review the Solved Examples on Literal Equations I shall keep updating the contents as time demands.
Comments, ideas, areas of improvement, questions, and constructive criticisms are welcome.
You may contact me.
If you are my student, please do not contact me here. Contact me via the school's system.
Thank you.

Samuel Dominic Chukwuemeka (Samdom For Peace) B.Eng., A.A.T, M.Ed., M.S

## Right Triangle

$perpendicular\:\:height = height \\[3ex] Area = \dfrac{1}{2} * base * height \\[5ex] height = \dfrac{2 * Area}{base} \\[5ex] base = \dfrac{2 * Area}{height} \\[5ex] hypotenuse^2 = height^2 + base^2...Pythagorean\:\:Theorem \\[3ex] hypotenuse = \sqrt{height^2 + base^2} \\[3ex] height = \sqrt{hypotenuse^2 - base^2} \\[3ex] base = \sqrt{hypotenuse^2 - height^2} \\[3ex] Perimeter = hypotenuse + height + base \\[3ex] Area = \dfrac{1}{2} * height * base * \sin (hypotenuseAngle) \\[5ex] Area = \dfrac{1}{2} * height * hypotenuse * \sin (baseAngle) \\[5ex] Area = \dfrac{1}{2} * base * hypotenuse * \sin (heightAngle) \\[5ex] Semiperimeter = \dfrac{height + base + hypotenuse}{2} \\[5ex] Semiperimeter - height = firstdifference \\[3ex] Semiperimeter - base = seconddifference \\[3ex] Semiperimeter - hypotenuse = thirddifference \\[3ex] Area = \sqrt{Semiperimeter * firstdifference * seconddifference * thirddifference}...Hero's\:\:Formula\:\:or\:\:Heron's\:\:Formula \\[5ex] hypotenuse = {Perimeter^2 - 4 * Area}{2 * Perimeter} \\[5ex] base = \dfrac{(Perimeter - hypotenuse) \pm Math.sqrt((hypotenuse - Perimeter)^2 - 8 * Area)}{2} \\[5ex] height = \dfrac{2 * Area}{base}$

## Triangle

$Perimeter = firstside + secondside + thirdside \\[5ex] Area = \dfrac{1}{2} * firstside * secondside * \sin (thirdAngle) \\[5ex] Area = \dfrac{1}{2} * firstside * thirdside * \sin (secondAngle) \\[5ex] Area = \dfrac{1}{2} * secondside * thirdside * \sin (firstAngle) \\[5ex] Semiperimeter = \dfrac{firstside + secondside + thirdside}{2} \\[5ex] Semiperimeter - firstside = firstdifference \\[3ex] Semiperimeter - secondside = seconddifference \\[3ex] Semiperimeter - thirdside = thirddifference \\[3ex] Area = \sqrt{Semiperimeter * firstdifference * seconddifference * thirddifference}...Hero's\:\:Formula\:\:or\:\:Heron's\:\:Formula \\[5ex] \underline{Cosine\:\:Law} \\[3ex] firstside^2 = secondside^2 + thirdside^2 - 2 * secondside * thirdside \cos (firstAngle) \\[3ex] secondside^2 = firstside^2 + thirdside^2 - 2 * firstside * thirdside \cos (secondAngle) \\[3ex] thirdside^2 = firstside^2 + secondside^2 - 2 * firstside * secondside \cos (thirdAngle)$

## Square

$side = length = width = height \\[3ex] Area = side^2 \\[3ex] side = \sqrt{Area} \\[3ex] Perimeter = 4 * side \\[3ex] side = \dfrac{Perimeter}{4} \\[5ex] diagonal = side * \sqrt{2} \\[3ex] side = \dfrac{diagonal * \sqrt{2}}{2} \\[5ex] Area = \dfrac{Perimeter^2}{16} \\[5ex] Perimeter = 4 * \sqrt{Area} \\[3ex] Area = \dfrac{diagonal^2}{2} \\[5ex] diagonal = \sqrt{2 * Area} \\[3ex] Perimeter = 2 * diagonal * \sqrt{2} \\[3ex] diagonal = \dfrac{Perimeter * \sqrt{2}}{4}$

## Circle

$Area = A \\[3ex] Circumference = C \\[3ex] Radius = r \\[3ex] Diameter = d \\[3ex] d = 2r \\[3ex] r = \dfrac{d}{2} \\[5ex] A = \pi r^2 \\[3ex] A = \dfrac{\pi d^2}{4} \\[5ex] C = 2\pi r \\[3ex] C = \pi d \\[3ex] r = \dfrac{\sqrt{A\pi}}{\pi} \\[5ex] r = \dfrac{C}{2\pi} \\[5ex] d = \dfrac{2\sqrt{A\pi}}{\pi} \\[5ex] r = \dfrac{C}{\pi} \\[5ex] A = \dfrac{C^2}{4\pi} \\[5ex] C = 2\sqrt{A\pi}$

## Cube

$edge = side = length = width = height \\[3ex] Surface\:\:Area = 6 * side^2 \\[3ex] side = \sqrt{\dfrac{Surface\:\:Area}{6}} \\[5ex] Volume = side^3 \\[3ex] side = \sqrt[3]{Volume} \\[3ex] Volume = \dfrac{side * Surface\:\: Area}{6} \\[5ex] side = \dfrac{6 * Volume}{Surface\:\:Area} \\[5ex] Surface\:\:Area = \dfrac{6 * Volume}{side} \\[5ex] Volume = \dfrac{Surface\:\:Area * \sqrt{6 * Surface\:\:Area}}{36} \\[5ex] side = \dfrac{diagonal * \sqrt{2}}{2} \\[5ex] diagonal = \sqrt{2} * side \\[3ex] Surface\:\:Area = 3 * diagonal^2 \\[3ex] diagonal = \dfrac{\sqrt{3 * Surface\:\:Area}}{3} \\[5ex] Volume = \dfrac{diagonal^3 * \sqrt{2}}{4} \\[5ex] diagonal = \sqrt[3]{2 * Volume * \sqrt{2}} \\[3ex] diagonal = \dfrac{1}{6} * \sqrt{\dfrac{72 * Volume}{side}} \\[5ex] Surface\:\:Area = \dfrac{12 * Volume}{diagonal} \\[5ex] Volume = \dfrac{Surface\:\:Area * diagonal}{12}$

## Right Cone

Curved Surface Area = Lateral Surface Area
Height = Perpendicular Height

$Volume\:\:of\:\:Cone = \dfrac{1}{3} * Volume\:\:of\:\:Cylinder \\[5ex] Lateral\:\:Surface\:\:Area = LSA \\[3ex] Base\:\:Area = BA \\[3ex] Total\:\:Surface\:\:Area = TSA \\[3ex] Volume = V \\[3ex] Diameter = d \\[3ex] Radius = r \\[3ex] Height = h \\[3ex] Slant Height = l \\[3ex] r = \dfrac{d}{2} \\[5ex] d = 2r \\[3ex] l = \sqrt{h^2 + r^2} \\[3ex] l = \dfrac{\sqrt{4h^2 + d^2}}{2} \\[5ex] h = \sqrt{l^2 - r^2} \\[3ex] h = \dfrac{\sqrt{4l^2 - d^2}}{2} \\[5ex] r = \sqrt{l^2 - h^2} \\[3ex] d = 2 * \sqrt{l^2 - h^2} \\[3ex] BA = \pi r^2 \\[3ex] r = \dfrac{\sqrt{BA * \pi}}{\pi} \\[5ex] BA = \dfrac{\pi d^2}{4} \\[5ex] d = \dfrac{2\sqrt{BA * \pi}}{\pi} \\[5ex] LSA = \pi rl \\[3ex] LSA = \dfrac{\pi dl}{2} \\[5ex] l = \dfrac{LSA}{\pi r} \\[5ex] LSA = \pi r\sqrt{h^2 + r^2} \\[3ex] h = \dfrac{\sqrt{LSA^2 - \pi^2 r^4}}{\pi r} \\[5ex] TSA = BA + LSA \\[3ex] TSA = \pi r(r + l) \\[3ex] l = \dfrac{TSA - \pi r^2}{\pi r} \\[5ex] TSA = \dfrac{\pi d(d + 2l)}{4} \\[5ex] l = \dfrac{4 * TSA - \pi d^2}{2\pi d} \\[5ex] r = \dfrac{-\pi l \pm \sqrt{\pi^2 l^2 + 4\pi * TSA}}{2\pi} \\[5ex] TSA = \pi r(r + \sqrt{h^2 + r^2}) \\[3ex] h = \dfrac{\sqrt{TSA(TSA - 2\pi r^2)}}{\pi r} \\[5ex] V = \dfrac{BA * h}{3} \\[5ex] V = \dfrac{\pi r^2h}{3} \\[5ex] V = \dfrac{\pi hd^2}{12} \\[5ex] V = \dfrac{\pi h(l^2 - h^2)}{3} \\[5ex] h = \dfrac{3V}{\pi r^2} \\[5ex] r = \dfrac{\sqrt{3V\pi h}}{\pi h}$

## Right Cylinder

Curved Surface Area = Lateral Surface Area
Height = Perpendicular Height

$Volume\:\:of\:\:Cylinder = 3 * Volume\:\:of\:\:Cone \\[3ex] Lateral\:\:Surface\:\:Area = LSA \\[3ex] Base\:\:Area = BA \\[3ex] Total\:\:Surface\:\:Area = TSA \\[3ex] Volume = V \\[3ex] Diameter = d \\[3ex] Radius = r \\[3ex] Height = h \\[3ex] r = \dfrac{d}{2} \\[5ex] d = 2r \\[3ex] LSA = 2\pi rh \\[3ex] r = \dfrac{LSA}{2\pi h} \\[5ex] h = \dfrac{LSA}{2\pi r} \\[5ex] LSA = \pi dh \\[3ex] h = \dfrac{LSA}{\pi d} \\[5ex] d = \dfrac{LSA}{\pi h} \\[5ex] BA = \pi r^2 \\[3ex] r = \dfrac{\sqrt{\pi BA}}{\pi} \\[5ex] r = \dfrac{1}{\pi} * \sqrt{\dfrac{\pi(TSA - 2 * LSA)}{2}} \\[5ex] BA = \dfrac{\pi d^2}{4} \\[5ex] d = \dfrac{2\sqrt{\pi BA}}{\pi} \\[5ex] d = \dfrac{\sqrt{2\pi (TSA - LSA)}}{\pi} \\[5ex] TSA = 2\pi r(r + h) \\[3ex] h = \dfrac{TSA - 2\pi r^2}{2\pi r} \\[5ex] r = \dfrac{-\pi h \pm \sqrt{\pi(\pi h^2 + 2 * TSA)}}{2\pi} \\[5ex] TSA = 2BA + LSA \\[3ex] BA = \dfrac{TSA - LSA}{2} \\[5ex] LSA = TSA - 2BA \\[3ex] TSA = \pi d\left(\dfrac{d + 2h}{2}\right) \\[5ex] h = \dfrac{2 * TSA - \pi d^2}{2\pi d} \\[5ex] d = \dfrac{-\pi h \pm \sqrt{\pi(h^2 + 2 * TSA)}}{\pi} \\[5ex] h = \dfrac{LSA * \sqrt{\pi * BA}}{\pi * BA} \\[5ex] h = \dfrac{LSA}{\sqrt{2\pi(TSA - LSA)}} \\[5ex] BA = \dfrac{LSA^2}{\pi h^2} \\[5ex] BA = \dfrac{(4 * TSA + \pi h^2) \pm h\sqrt{\pi(\pi h^2 - 8 * TSA)}}{8} \\[5ex] LSA = h\sqrt{BA * \pi} \\[3ex] LSA = \dfrac{-\pi h^2 \pm h\sqrt{\pi(\pi h^2 + 8 * TSA)}}{4} \\[5ex] TSA = 2 * BA \pm h\sqrt{\pi * BA} \\[3ex] TSA = \dfrac{LSA(2 * LSA + \pi h^2)}{\pi h^2} \\[5ex] V = \pi r^2h \\[3ex] r = \dfrac{2V}{LSA} \\[5ex] d = \dfrac{4V}{LSA} \\[5ex] r = \dfrac{\sqrt{Vh\pi}}{h\pi} \\[5ex] V = BA * h \\[3ex] BA = \dfrac{V}{h} \\[5ex] h = \dfrac{V}{BA} \\[5ex] h = \dfrac{V}{\pi r^2} \\[5ex] h = \dfrac{4V}{\pi d^2} \\[5ex] V = \dfrac{\pi d^2h}{4} \\[5ex] d = \dfrac{\sqrt{Vh\pi}}2{h\pi} \\[5ex] V = \dfrac{LSA^2}{h\pi} \\[5ex] LSA = \sqrt{Vh\pi} \\[3ex] h = \dfrac{LSA^2}{4V\pi} \\[5ex] V = \dfrac{(h^3\pi + 4 * TSA * h) \pm h^2\sqrt{\pi(h^2\pi + 8 * TSA)}}{8} \\[5ex] TSA = \dfrac{2V + h\sqrt{Vh\pi}}{h} \\[5ex] TSA = \dfrac{2V + 2\pi rh^2}{h} \\[5ex] r = \dfrac{TSA * h - 2V}{2\pi h^2} \\[5ex] d = \dfrac{TSA * h - 2V}{\pi h^2} \\[5ex] h = \dfrac{TSA \pm \sqrt{TSA^2 - 16\pi rV}}{4\pi r}$