How to measure the focal length of a lens

Your zoom lens may have a focal length range of 28-200mm and a focusing range from perhaps 0.5m to infinity. But you probably do not have the full range of focal lengths available when focused fairly close e.g. 4m. I recently measured the focal length of a “28-200mm” zoom from a well respected manufacturer and found that when focused at 4m it was 27.5mm to 153mm.

So, how do we measure the focal length? You will need, in addition to the lens and camera, a tripod, a 5m measuring tape and a brick wall or other suitable target. Some masking tape will also be useful.

The method:

Set up the camera on the tripod about 4 to 5 metres from a brick wall so that the camera is square to the surface of the wall. With the camera vertical (portrait orientation) focus the camera and stick a small piece of tape on the wall in the centre of the field of view (this is so that we can identify the bricks later) and take a picture.

View the picture on your monitor and you will see that the field of view will cover (perhaps) six courses of bricks. Note that for a wide angle lens, you might need a larger wall or move the camera nearer and use it horizontal (landscape orientation).

We need to make just two measurements:

1. Measure the distance from the lens to the wall – this is the “object distance”. Ideally this should be from the front nodal point of the lens to the wall. For now, let’s assume that the front nodal point is about 20mm behind the front element of the lens. (If you want to be really accurate you could measure from the lens mount and determine the position of the nodal point later).

2. Measure the “size of object”. this is the distance measured on the wall corresponding to the field of view in the image. You will be able to measure this very accurately (not just “6 bricks”) by identifying detail in the texture of the bricks.

To calculate the focal length:

We first need to calculate the magnification, m. or rather, its reciprocal, 1/m.

m = (size of image)÷(size of object)  so  1/m = (size of object)÷(size of image)

The “size of image” is the size of the sensor in the camera. You should be able to find this in your camera manual. In my case this is 23.5mm (on the longest side). The “size of object” measured 612mm, so

1/m = 612÷23.5 = 26.043

Knowing the object distance, u, in this case 4150mm, we can calculate the focal length, f, from:

f = u÷(1+1/m) = 4150÷(1+26.043) = 4150÷27.043 = 153.5

Focal length is 153.5mm.

Accuracy

We made just two measurements. The accuracy of each of these should be within ± 5mm. Perhaps we should be pessimistic and say that the error in u may be ± 25mm to allow for estimating the position of the nodal point.

4150 ± 25 and 612 ± 5, both of these are within ±1%. We do not know how accurate the manufacturer’s stated sensor size is so we must accept the value quoted (23.5mm). Overall we can expect a result well within ±2%.

Comment

I was a little surprised and disappointed to find that my zoom lens (manufactured about 2003) has a maximum focal length of 153mm when focused at this distance, so to check my method I measured the focal length of an smc PENTAX-M 1:4 200mm. – a prime lens manufactured about 1975-80. The result was 201.7mm, again with an accuracy well within ±2%. It was obvious, looking through the viewfinder, that the magnification of the zoom lens was only about three quarters that of the prime lens, so the measured result came as no surprise. However, comparing the apparent magnification of the two lenses (through the viewfinder) when viewing a distant scene (effectively at infinity) they appeared very similar, suggesting that in this case the zoom has a focal length of about 200mm.

I suspect that this effect is common to all wide range zoom lenses, and is a necessary compromise in the design process.

Footnote: More technical stuff

The focal length calculation is based on the simple formula:

1/f = 1/u + 1/v

where f is focal length, u is object distance and v is image distance. In theoretical optics the formula applies to a “thin” lens (i.e. a single element of negligible thickness) but can be applied to real multi-element lenses provided we measure distances from the appropriate nodal point of the lens. The real obstacle is that we cannot measure v, the image distance, because it is inside the camera. We can, however, measure the magnification, m, and make use of the formula:

m = v/u

By combining these two formulae we obtain an expression for f in terms of u and m:

f = u÷(1+1/m)

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