Visual Angle

Background:

The subtended visual angle of an object is the angle formed by rays projecting from the eye to the top and bottom (or left and right sides) of an object. Visual angles are used to indicate the size of the retinal image of the object -- the larger the visual angle, the larger the retinal image size is.

The visual angle is influenced by two things -- the size of the object and the distance of the object from the eye. Bigger objects cast larger images on the retina than smaller objects. Thus, the larger the object is, the larger its visual angle will be. Closer objects cast larger images on the retina than smaller objects. Thus, the closer the object is to the eye, the larger its visual angle will be.

Calculating the visual angle is easy if you know the object's size and distance. You simply substitute those values into the following formula:

$$\large Visual\ Angle = 2 \cdot atan(\frac{\frac{Object\ Size} {2}} {Object\ Distance})\\$$

Where atan is the arc tangent (or inverse tangent) function which is often represented as tan-1 on calculators.

Visual angles are often reported in degrees (°), minutes (') and seconds ('') of visual arc. If you divide a circle into 360 wedges of equal size, each wedge would subtend 1°. If you divide a degree of arc into 60 equally sized wedges, each wedge would subtend 1 minute (') of arc. If you divide a minute of arc into 60 equally sized wedges, each wedge would subtend 1 second ('') of arc. Thus, 1'' of arc equals 1 / 3600 of a degree (1 / (60 * 60)). While that may seem like a very small angle, we need that degree of precision to measure some types of visual acuity. Detection acuity -- the smallest object that we can detect -- is approximately 0.5'' of subtended visual angle. Vernier acuity -- deciding if two lines are colinear (such as lining up the markings on a dial of a padlock with the markings around the dial) -- is approximately 1 to 2'' of subtended visual angle.

The Activity:

In the following activity, use the two sliders to manipulate the distance of the object (the cat) from the eye and the size of the object. The angle between the two lines projecting from the eye to the top and bottom of the cat is the visual angle. The calculated visual angle is shown in degrees (and fractional degrees) and in degrees, minutes and seconds of visual arc. Also shown is the formula and calculations from which the visual angle is derived.

• Increase the size of the cat and observe how the visual angle changes.
• Decrease the size of the cat and observe how the visual angle changes.
• Increase the distance of the cat from the eye and observe how the visual angle changes.
• Decrease the distance of the cat from the eye and observe how the visual angle changes.

Visual angle =

Distance: 50 400
Current distance =

Obj. Size: 25 300
Current obj. size =

Calculations:

Note: If you think that the ray from the eye to the bottom of the cat should touch the cat's front paw, you are wrong. In this example, the rays need to project to the highest part of the cat (the tip of its pinna [ear]) and to the lowest part of the cat (which is part of the cat's tail.) That is, part of the cat's tail is lower than its front paw. The ray appropriately projects to the Y-axis coordinate of the lowest part of tail.

If you are using a calculator to check the results, be sure that your calculator is returning degrees from the arc tangent function. Often calculators default to returning radians. To convert radians to degrees, multiply by 180 / π (approximately 57.296).

Visual Angle Calculator