Calculation of aperture

The f-number scale increases by multiples of

$\sqrt{2}$

(approximately 1.414) because doubling or halving the light entering the camera requires doubling or halving the area of the circular aperture, not its diameter. 

Here is the simple mathematical breakdown of why this happens: 

1. Light and Area

The amount of light that passes through a lens depends entirely on the area (A) of the optical opening. Since the aperture is a circle, its area is calculated using the formula:

 

This shows that the area (

$A$

) is proportional to the square of the diameter (

$D^2$

).

2. The Inverse Relationship

From your previous formula, the diameter is

. This means the f-number (

$N$

) is in the denominator. 

• To let in half the light, you need half the area.
• To get half the area, you must reduce the diameter by a factor of
  

  $\sqrt{2}$
  .
• Because
  

  $N$
  is inversely proportional to
  

  $D$
  , you must multiply the f-number by
  

  $\sqrt{2}$
  to make the opening smaller by that exact ratio.

Google Gemini

Let 焦距 (f) be 35 mm and F值 (N) = 2

光圈孔直徑 (D) = f / N

D = 35 mm ÷ 2 = 17.5 mm

A = 3.1416 * 8.75mm * 8.75mm = 240.5 mm square

0.5 A = 3.1416  * r * r = 120.3 mm square

r * r = 38.3 mm square

r = 6.2 mm

Original diameter is 17.5mm

Area in half, new diameter is 12.4mm

D = f / N

12.4 = 35 / N

N = 2.8