How to calculate the area of an opening?

The area of the opening of a circular hole is A, which is equal to pi multiplied by the square of the radius. It can also be expressed as pi multiplied by the square of the diameter, divided by 4. For a camera lens, the diameter D can be calculated from the focal length f and the aperture number N: D = f/N. Therefore, for the area of the opening: A = pi multiplied by the square of (f/(2N)).

How does a telescope's aperture affect its ability to collect light?

The light-gathering efficiency is proportional to the area of the aperture. Since the area is proportional to the square of the diameter, a telescope with twice the diameter of another will gather four times as much light. A 200 mm telescope gathers about 800 times more light than the human eye. For comparison, the human pupil has a diameter of 7 mm. This is why large telescopes can observe dark objects far away in space.

How big is the aperture area of a camera lens with a focal length of 50 mm and an f/1.8?

The entrance pupil diameter is approximately 27.8 mm and is calculated by dividing 50 by 1.8. The area is the result of pi multiplied by the square of the value that results from 27.8 divided by 2, which is approximately 606 square millimeters or 6.06 square centimeters. High aperture lenses such as f/1.4 or f/1.8 are often preferred for low-light photography because they have a significantly larger area of the diaphragm than lower aperture lenses such as f/4 or f/5.6 at the same focal length.

Why does the amount of light gathered quadruple when the diameter of the aperture is doubled?

The area is determined by the square of the diameter. If the diameter is doubled, both the area and the amount of light gathered are quadrupled. If the diameter is tripled, the amount of light gathered is nine times greater. This quadratic relationship is why lenses with large diameters are much more effective at gathering light than lenses with small diameters.

What is the formula for f-number with respect to aperture diameter?

The F-number, also called the focal ratio, is calculated as N = f / D where f is the focal length and D is the diameter of the aperture. Rearranging this gives D = f / N. So for a lens with 100 mm focal length set to f/2.8, the diameter would be approximately 35.7 mm while the same diameter on the same lens set to f/5.6 would be approximately 17.9 mm.

Aperture Area Calculator

Enter what you know

Aperture diameter (D)

Focal length (f)

f-number (N)

Aperture area (A)

Compare the light-gathering power

Enter a reference aperture to see how many times more light this one collects.

Enter an aperture diameter to see its area. For a camera lens, enter the focal length and the f-number instead, and the diameter and area fill in for you. You can also enter an area to work back to the diameter.

Please input values into the fields

Analysis

This Aperture Area Calculator allows you to calculate the area of openings through which light enters in optical systems, telescopes, microscopes, camera lenses and antennas. Enter known data for any one parameter. If you enter the diameter, the area itself or the focal length and f-number of a camera lens, the remaining results are automatically calculated. The area is the most important result as it determines the light gathering properties of the device.

What is an opening area?

The aperture is the hole or opening through which light passes. Its area corresponds to the size of a cross-section of this circular hole when cut in a plane and is expressed in square units such as square millimeters or centimeters.

The aperture area is important because it is proportional to the amount of light that an instrument can collect. The larger the aperture area, the more light can be gathered. This is why large telescopes can see faint galaxies that cannot be seen with small telescopes. Also, a larger aperture in camera lenses allows pictures to be taken even in dark rooms. Because the area is determined by the square of the diameter, even a slight change in the diameter leads to a significant change in light gathering ability.

How to use this calculator:

Enter a value and select the appropriate unit. For example, if you enter the diameter of a telescope as 200 mm, the area will be immediately shown. With camera lenses leave the diameter blank and enter focal length and f-number. The calculator first calculates the entrance pupil diameter and then the area. You can also enter an area to calculate the corresponding diameter.

To compare different devices open the aperture option and enter the reference diameter. The calculation tool will show you by what factor your opening collects more light than the reference diameter.

Formula for calculating the opening area:

Since the opening is circular in shape, its area can be calculated by multiplying pi with the square of the radius. Since the radius is half of the diameter, the area can be expressed directly in terms of the diameter as follows:

A = \pi \left(\frac{D}{2}\right)^2 = \frac{\pi D^2}{4}

In this formula A is the area of the opening, D is the diameter of the opening and π is approximately 3.14159. To calculate the diameter from the area, divide by π then take the square root.

D = 2\sqrt{\frac{A}{\pi}}

Aperture value (f-number) and aperture diameter:

In camera lenses, the aperture is seldom measured directly. Usually only the focal length and the aperture value (also called F-number) are known. The aperture value N is the result of dividing the focal length f by the diameter of the aperture opening D.

N = \frac{f}{D} \qquad \Longrightarrow \qquad D = \frac{f}{N}

If you put this diameter into the formula for area, then you can calculate the aperture area directly from focal length and f-number.

A = \pi \left(\frac{f}{2N}\right)^2

Typical f-numbers are e.g. 1.4, 2, 2.8, 4, 5.6. With each stop decrease in aperture the f-number is multiplied by the square root of two while the diameter is divided by the square root of two. This halves the area of the aperture opening. That's why when you decrease the aperture by one stop, the amount of light falling on the sensor is halved.

Example calculation:

Let's take an example of a standard lens with a focal length of 70mm and f-number of 1.4. First we calculate the entrance pupil diameter.

D = \frac{f}{N} = \frac{70}{1.4} = 50 \ \text{mm}

Then we square the radius of 25 mm and multiply it by pi.

A = \pi \times 25^2 = 625\pi \approx 1963.5 \ \text{mm}^2

So the aperture area is about 1963.5 square millimeters, or 19.64 square centimeters. If you enter 70 in the focal length field and 1.4 in the f-number field, the calculator shows the same result along with the diameter and radius.

Overview of typical opening diameters

The table shows the area of some common aperture diameters rounded to a reasonable number of places. Note that the area increases rapidly with increasing diameter as it is squared.

Aperture	Diameter	Area
Dark-adapted eye pupil	7 mm	38.5 sq mm
50 mm f/1.8 camera lens	27.8 mm	606 sq mm
100 mm refractor	100 mm	7854 sq mm
200 mm telescope	200 mm	31416 sq mm
8-inch Dobsonian	203.2 mm	32429 sq mm

Openings and telescopes

Since area is proportional to the square of the diameter, a telescope with twice the diameter of another will collect four times as much light, while one three times the diameter will collect nine times as much. The aperture area of a 200 mm telescope is about 31400 square millimeters, corresponding to an effective collecting area of about 800 times that of the human eye's 7 mm pupil. This collects about 800 times more light, allowing very faint objects invisible to the naked eye to be seen.

To compare two devices quickly, you can simply divide their areas. This is equivalent to squaring the ratio of their diameters. The option for calculating light collection efficiency in this tool calculates the value exactly this way.

Opening in photography

In a camera the aperture is one side of the exposure triangle, with the other two sides being shutter speed and ISO. With larger apertures, or small f-numbers like f/1.4 or f/1.8, you get more area open up to let in light so that you can shoot in darker places without increasing your ISO and therefore adding noise. The depth of field also decreases which will lead to a blurred background.

Two lenses with the same aperture will let in the same amount of light onto the sensor (relative to the area of the sensor) at different focal lengths, but lenses with longer focal length have a larger physical aperture size. This is why the front lens elements on a 400 mm f/5.6 lens are much bigger than those on a 50 mm f/5.6 lens.

Applications of the shutter surface

Aperture is used wherever light or radiation needs to be gathered or shaped. Astronomers determine the size of mirrors and lenses in telescopes so that they can observe dimmer celestial objects. Photographers balance aperture, shutter speed, and ISO. Microscopists relate aperture to resolution. Even antenna designers discuss effective aperture, which determines how much signal a parabolic dish can receive. This quantity is not calculated from an actual circle, however; it is derived from the gain and wavelength.

Tips for accurate results

The aperture should be measured as the widest unobstructed opening, not including any outer housing. Be sure to use consistent units. If the diameter is given in millimeters, then the area will be in square millimeters. Note that area increases proportionally to the square of the diameter, so small errors in the diameter can lead to even larger errors in the area and thus estimates of light gathering ability.

This tool is for general education and planning purposes only. If exact optical design or device specification is required, please verify these values with the manufacturer data and applicable standards for your application.

Frequently asked questions

How to calculate the area of an opening?: The area of the opening of a circular hole is A, which is equal to pi multiplied by the square of the radius. It can also be expressed as pi multiplied by the square of the diameter, divided by 4. For a camera lens, the diameter D can be calculated from the focal length f and the aperture number N: D = f/N. Therefore, for the area of the opening: A = pi multiplied by the square of (f/(2N)).
How does a telescope's aperture affect its ability to collect light?: The light-gathering efficiency is proportional to the area of the aperture. Since the area is proportional to the square of the diameter, a telescope with twice the diameter of another will gather four times as much light. A 200 mm telescope gathers about 800 times more light than the human eye. For comparison, the human pupil has a diameter of 7 mm. This is why large telescopes can observe dark objects far away in space.
How big is the aperture area of a camera lens with a focal length of 50 mm and an f/1.8?: The entrance pupil diameter is approximately 27.8 mm and is calculated by dividing 50 by 1.8. The area is the result of pi multiplied by the square of the value that results from 27.8 divided by 2, which is approximately 606 square millimeters or 6.06 square centimeters. High aperture lenses such as f/1.4 or f/1.8 are often preferred for low-light photography because they have a significantly larger area of the diaphragm than lower aperture lenses such as f/4 or f/5.6 at the same focal length.
Why does the amount of light gathered quadruple when the diameter of the aperture is doubled?: The area is determined by the square of the diameter. If the diameter is doubled, both the area and the amount of light gathered are quadrupled. If the diameter is tripled, the amount of light gathered is nine times greater. This quadratic relationship is why lenses with large diameters are much more effective at gathering light than lenses with small diameters.
What is the formula for f-number with respect to aperture diameter?: The F-number, also called the focal ratio, is calculated as N = f / D where f is the focal length and D is the diameter of the aperture. Rearranging this gives D = f / N. So for a lens with 100 mm focal length set to f/2.8, the diameter would be approximately 35.7 mm while the same diameter on the same lens set to f/5.6 would be approximately 17.9 mm.

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Wikipedia: Aperture
The aperture in optics and photography, and how it controls light.
Wikipedia: F-number
Definition of the f-number, the f-stop scale, and the entrance pupil.
Wikipedia: Optical telescope (light-gathering power)
How aperture area sets the light-gathering power of a telescope.