Determining the equation of the volume of a cylinder

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SUMMARY

The equation for the volume of a cylinder, expressed solely in terms of the radius (r), is derived from the relationship between the height (h) and the diameter (D) of an equilateral cylinder. In this case, the diameter is equal to the height, leading to the equation D = h = 2r. Consequently, the volume formula becomes V(r) = π(r²)(2r) = 2πr³. This derivation clarifies how to express the volume without directly referencing height or length.

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  • Understanding of basic geometry concepts, specifically cylinders.
  • Familiarity with mathematical constants, particularly π (pi).
  • Knowledge of algebraic manipulation to derive equations.
  • Basic understanding of the relationship between radius, diameter, and height in cylindrical shapes.
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  • Study the properties of geometric shapes, focusing on cylinders and their volume calculations.
  • Explore the derivation of volume formulas for other geometric solids, such as cones and spheres.
  • Learn about the applications of volume calculations in real-world scenarios, such as engineering and architecture.
  • Investigate the significance of π in various mathematical contexts beyond geometry.
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Students, educators, and anyone interested in understanding geometric volume calculations, particularly those focusing on cylindrical shapes.

bwell
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Hi,

My knowledge is not too good and I'm trying to help my son find out how we would go about determining the equation of the volume of the cyclinder V(r), expressed in terms of r alone? We do not know how we would go about this without using l or h?

Any help would be really appreciative (Smile)
 
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bwell said:
Hi,

My knowledge is not too good and I'm trying to help my son find out how we would go about determining the equation of the volume of the cyclinder V(r), expressed in terms of r alone? We do not know how we would go about this without using l or h?

Any help would be really appreciative (Smile)

Since the volume of a cylinder depends both on the radius of the base and its height, either you have misunderstood what you are being asked to do, or have not posted the complete question.

Post the question exactly as worded and any other contextual material which sets up the background for the question.

CB
 
If the cylinder is of an equilateral one, then its diameter is equal to its height, i.e. D=h. But D=2r, so we have h=2r.
Then, the volume of the equilateral cylinder, V(r)=pi(r^2)h=pi(r^2)(2r)=2pir^3.
 

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