If there is one prayer that you should pray/sing every day and every hour, it is the LORD's prayer (Our FATHER in Heaven prayer)
It is the most powerful prayer. A pure heart, a clean mind, and a clear conscience is necessary for it.
- Samuel Dominic Chukwuemeka

"When JESUS saw the crowds, He went up on the mountain and sat down. His disciples came to Him, and He began to teach them, saying: "Blessed are the poor in spirit, for theirs is the kingdom of heaven. Blessed are those who mourn, for they will be comforted. Blessed are the meek, for they will inherit the earth. Blessed are those who hunger and thirst for righteousness, for they will be filled. Blessed are the merciful, for they will be shown mercy. Blessed are the pure in heart, for they will see GOD. Blessed are the peacemakers, for they will be called sons of GOD. Blessed are those who are persecuted because of righteousness, for theirs is the kingdom of heaven. Blessed are you when people insult you, persecute you, and falsely say all kinds of evil against you because of Me. Rejoice and be glad, because great is your reward in heaven; for in the same way they persecuted the prophets before you." - Matthew 5:1 - 12

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

Mensuration Formulas

Samuel Dominic Chukwuemeka Good morning.
I greet you this day, Wed May 14 2025 03:09:16 GMT+0000 (Coordinated Universal Time).
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) \\[5ex] firstAngle = \cos^{-1} \left(\dfrac{secondside^2 + thirdside^2 - firstside^2}{2 * secondside * thirdside}\right) \\[5ex] secondAngle = \cos^{-1} \left(\dfrac{firstside^2 + thirdside^2 - secondside^2}{2 * firstside * thirdside}\right) \\[5ex] thirdAngle = \cos^{-1} \left(\dfrac{firstside^2 + secondside^2 - thirdside^2}{2 * firstside * secondside}\right) $





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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} $





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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} $





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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} $





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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} $





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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} $

Oblique Cylinder





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