# How to calculate wheel radius

The basic vehicle dynamics parameters calculations, like wheel torque or wheel speed, are using the wheel radius as input. Using the size markings of any tire, we can calculate its radius. To recall what’s the meaning of the tire markings, read the article Vehicle tire markings explained.

Using the size marking of the tire we can calculate the free static wheel radius. Free means that there is no load applied on the wheel and static means that the wheel is stationary.

The main size marking of a tire is defined as:

TireWidth [mm] / AspectRatio [%] ConstructionType RimDiameter [in]

For example, a tire with the size marking 225/55R17 has:

• the width of 225 mm
• the aspect ration of 55 %
• radial plies (layers)
• interior (rim) diameter of 17 inch

The aspect ratio defines the relationship between tire width and sidewall height (excluding rim diameter).

The aspect ratio (AR) is defined in percentage, as the ratio between tire sidewall height and tire width.

$\text{AR [%]}=\frac{\text{H [mm]}}{\text{W [mm]}} \cdot 100 \tag{1}$

where:

AR – aspect ratio
H – tire side wall height
W – tire width
D – rim diameter

In order to calculate the diameter of the wheel, we need to know:

• rim diameter (taken from size marking)
• tire sidewall height (calculated from aspect ratio)

From (1) we can calculate the tire side wall height H [mm]:

$H = \frac{AR \cdot W}{100} \tag{2}$

The wheel diameter dw [mm] is the sum between the rim diameter and twice the height of the tire:

$d_{w} = D + 2 \cdot H \tag{3}$

The (free static) wheel radius rw [mm] is half the wheel diameter:

$r_{w} = \frac{d_{w}}{2} \tag{4}$

To take into account the deformation of the tire, we can apply a coefficient of 0.95, which means that the free static radius will decrease with approx. 5% under load (vehicle weight). In this case the static radius of the wheel will be:

$r_{w} = 0.95 \cdot \frac{d_{w}}{2}$

If needed, we can also calculate the circumference of the wheel C [mm], as:

$C = \pi \cdot d_{w} \tag{5}$

Example. Calculate the wheel diameter [mm], radius [mm] and circumference [mm] for the tire with the size marking 225/55R17.

Step 1. Write down the known parameters and convert rim diameters from inch in mm.

$\begin{split} W &= 225 \text{ mm}\\ AR &= 55 \text{ %}\\ D &= 17 \text{ in} = 17 \cdot 25.4 = 431.8 \text{ mm} \end{split}$

Step 2. Calculate the tire sidewall height.

$H = \frac{AR \cdot W}{100} = \frac{55 \cdot 225}{100} = 123.75 \text{ mm}$

Step 3. Calculate the wheel diameter

$d_{w} = D + 2 \cdot H = 431.8 + 2 \cdot 123.75 = 679.3 \text{ mm}$

Step 4. Calculate the wheel radius

$r_{w} = \frac{d_{w}}{2} = \frac{679.3}{2} = 339.65 \text{ mm}$

Step 5. Calculate the wheel circumference

$C = \pi \cdot d_{w} = 3.1416 \cdot 679.3 = 2134.0889 \text{ mm}$

The steps for wheel radius calculation can be summarized in a Scilab script.

clc
sizeMarking = "225/55R17";
W = eval(part(sizeMarking,1:3));
AR = eval(part(sizeMarking,5:6));
D = eval(part(sizeMarking,8:9));

H = W * AR / 100;
dw = D*25.4 + 2*H;
rw = dw/2;
C = %pi*dw;

mprintf("\ndw = %.2f mm\nrw = %.2f mm\nC = %.2f mm\n",dw,rw,C)


Executing the script will output the following results in the Scilab console:

dw = 679.30 mm
rw = 339.65 mm
C = 2134.08 mm

To use the script for different tire size markings, change the sizeMarking variable with the corresponding string and execute the script. The results will be output in the Scilab console. For example, for sizeMarking = "255/60R18"; we get:

dw = 763.20 mm
rw = 381.60 mm
C = 2397.66 mm

You can also check your results using the calculator below.

### Wheel radius calculator

 W [mm] AR [%] D [in] / R rw [mm] = dw [mm] = rw [cm] = dw [cm] = rw [m] = dw [m] =

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