CSE 111: Programming with Functions

W01 Formula: Approximate Volume of a Tire

Let’s simplify the volume inside a tire to two cylinders as follows:

  1. an inner cylinder that is formed by the wheel (or rim as some people call it)
  2. an outer cylinder that contains both the tire and the wheel

To obtain the volume inside the tire only, we subtract the volume of the inner cylinder from the volume of the outer cylinder.

volumetire  ≅  volumeouter  −  volumewheel

We know the formula for the volume of a cylinder is

v = πr2h

where v is the volume, r is the radius, and h is the height of the cylinder.

Now let’s consider the three numbers we have for a tire in the United States. For example, 185/70R14.

  1. The first number is the width of the tire in millimeters, which we will call W.
  2. The second number is the aspect ratio of the tire, which we will call A. The height H of the sidewall of a tire is computed with this formula:
    H =
    A
    100
    W
  3. The third number is the diameter of the wheel in inches, which we will call D. Unfortunately, the diameter is in inches, but we want to use millimeters. To convert D from inches to millimeters we multiply it by 25.4. Also, we need the radius of the wheel and not its diameter, so we divide the diameter by 2. This means the formula for the radius of the wheel is:
    rwheel =
    25.4 D
    2

Starting with our first formula, we can now derive a formula that approximates the volume of air in a tire.

volumetire  ≅  volumeouter  −  volumewheel
volumetire  ≅  π r
2
outer
h  −  π r
2
wheel
h
volumetire  ≅  π h
r
2
outer
 −  r
2
wheel
volumetire  ≅  π W
(rwheel + H)2  −  r
2
wheel
volumetire  ≅  π W
(r
2
wheel
 +  2 rwheel H  +  H2)  −  r
2
wheel
volumetire  ≅  π W
2 rwheel H  +  H2
volumetire  ≅  π W H
2 rwheel  +  H
volumetire  ≅  π W 
A
100
W
2 
25.4 D
2
 + 
A
100
W
volumetire  ≅ 
π W2 A
100
25.4 D  + 
W A
100
volumetire  ≅ 
π W2 A
100
100 (25.4 D)
100
 + 
W A
100
volumetire  ≅ 
π W2 A
100
2540 D  +  W A
100
volumetire  ≅ 
π W2 A
2540 D  +  W A
10,000
[cubic millimeters]

However, we want the answer expressed in liters, not cubic millimeters, so we divide by 1,000,000. Therefore, the volume in liters of a tire is approximated by this formula:

volumetire  ≅ 
π W2 A
2540 D  +  W A
10,000,000,000
[liters]

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