Resources: Worksheet, WolframAlphaGoogle Maps, Tape Measures, Meter Sticks, Rulers, Paper, Tape, Paper Clips, Binder Clips 

Terminology: Small Angle Formula, Percent Error Formula

Whenever you look at an object, you are measuring its angular size - the amount of space it takes up in your field of view in degrees, minutes (1/60 of 1 degree), and seconds (1/60 of 1 minute or 1/3600 of 1 degree). Or you might quantify angular size in radians if you're mathematically-inclined. You can't directly measure an object's size in centimeters or inches unless you walk up to it and use a ruler. You know that objects far away look small and nearby objects look big, so when looking at an object, your brain combines the angular size of the object with your guess of its distance to give you an idea of its actual size.

Humans have evolved binocular vision to help us make these distance guesses for things that might affect our survival (e.g. lions, tigers, and bears -- oh my!). To figure out the distance to something, our brains also use the apparent sizes of well-known objects nearby to the object of interest, such as buildings and trees in the object's vicinity. In Astronomy, sizes are more uncertain; objects that appear close to one another in the sky may in fact be many light-years apart. Our basic measurement of size in Astronomy is angular size.

In order to learn the true physical size of an object, we must first record the angular size and find the distance to the object by some independent method. Conversely, if the physical size of an object is known, this can be combined with its apparent angular size to determine its distance. In these relationships, a third desired quantity can always be calculated if two quantities are known; this is done by using the Small Angle Formula.

Parallax with hands

Complex and precise instruments exist and can be constructed for measuring the angular size of objects, but a set of rough measurement tools can be found at the end of most people's arms. Because humans are built to mostly the same proportions, if you hold your arms outstretched with your palms facing forward, your hands will have about the same angular size in your field of vision regardless of whether you are tall, short, big, or small. Your fingers and knuckles can be used to make rough measurements of angular sizes and distances on the sky as shown in the diagram to the right.

Other useful angular size rulers exist as well. For example, the full Moon is almost exactly one-half degree in extent as viewed from the surface of the Earth. Certain constellations and asterisms stretch across specific amounts of degrees of the sky.


Five hands extended at arm's length show how the finger 5 pinky, fingers 2-4 resting together, a fist, fingers 2 and 5 in the 'rock on' sign, and fingers 1 and 5 in the 'hang loose' sign can measure angular sizes of 1, 5, 10, 15, and 25, respectively. The diagram also shows that the Big Dipper asterism spans the fingers 1 and 5 25 degree measurement.

Learning Goals: Students will explore how an object's distance, physical size, and angular size relate to one another, and learn how astronomers use this relationship to determine the sizes of distant objects. Throughout the lab, using only the provided materials and their understanding of angular size, students will develop a method for estimating the diameter of the Old Capital building's dome with their lab groups.