Getting an Actual Size from a Measured Angular Size
This activity is based on the original one written by Professor Woody Sullivan, University of Washington.
Background
We know that the angular diameter of the Moon is about 0.5° (its angular size changes because its orbit is not a circle). Given the angular diameter of the Moon, we could use it to figure out such things as the angular size of a building or the length of a bridge located in front of the Moon from our viewpoint.
Since the Moon isn’t always available to use as a comparison in measuring angular sizes, here are a few handy rules that you should remember. With your arm outstretched, one finger-width is ~2° (little finger ~1°), three fingers are ~5°, one fist-width ~10°, index-to-little finger ~15°, and the shaka sign (hang loose) ~25°. These are shown in Figure 1.
Figure 1. Using your fingers and a hand to estimate angular sizes. (https://chandra.harvard.edu/graphics/photo/cosmic_distance.jpg)
This technique works because an angle a depends on the ratio of the linear size d of the object to the distance D to the object, as sketched in Figure 2. Since the ratio of finger size to arm size is approximately the same for different persons, most people have a finger-width of ~2° no matter what the length of their arms. But it is still best if you calibrate your own body using the Moon and objects that you have measured with your cross staff, which we will be doing in this activity.
Figure 2. The angular size of an object, a, is proportional to its actual size, d, divided by its distance, D.
Question: How many Moons could you line up, side by side, to extend from the horizon to the zenith? Picture a total eclipse of the Sun. What does this imply about the angular size of the Sun?
Learning Goals
- Build and correctly calibrate a cross staff.
- Accurately follow the directions for taking measurements.
- Complete the table of data, finding the average of the building height.
- Correctly calculate the uncertainty in the measurement.
- Extend your learning to a different object, stating why the knowledge of its actual size would be important.
Part 1 — Building the Cross Staff
The cross staff is a simple device that was used for centuries before ~ CE 1600 for measuring the angular size of a single object and thereby deriving its linear size or distance. It consists of a meterstick and a crosspiece attached at a right angle. Construct a cross staff using the crosspiece pattern on the last page of this activity and a meter stick, as shown in Figure 3. Then, follow these instructions carefully.
Materials needed: good sharp, pointy scissors; razor or glue; card stock; meter stick.
Steps for construction:
- Use the pattern on the last page of this exercise (see Figure 5) and carefully cut along the heavy outside lines.
- Make sure the entire surface of the back of the pattern is covered with glue. Carefully place the pattern on the card stock and press all surfaces to seal.
- Cut out the crosspiece using the pattern.
- Cut out the two small rectangular regions near the center of the crosspiece. These will be used to slide the crosspiece onto the meter stick to finish your cross staff. Use a razor or X-Acto knife for precise, sharp edges.
- Fold your crosspiece along the dashed lines so the pattern page with the centimeter marks is on the outside. To do this, scratch one tip of your scissors along the lines on the back side of the crosspiece using the meter stick to guide the scissors.
- Glue the pieces of your crosspiece together and make sure it slides properly onto your meter stick.
Figure 3. A simple cross staff made from readily available materials.
Part 2 — Calibrating your measurements
In the classroom, your instructor will instruct you to measure the angular size of a globe from different distances in the room.
- How do you expect the angular size a will vary with your distance D from the globe having a diameter d? Use the approximate formula to aid in your prediction: a is proportional to (d / D).
As part of this exercise, you must first determine the length of your step (stride) using centimeters. Count the number of steps you take to leisurely walk 1000 centimeters (10 meters). Divide the 1000 cm by the number of steps to calculate the length of your step in centimeters. There will necessarily be an uncertainty in this, but we address these in our tips below.
- How many steps did you take to go 1000 centimeters (10 meters)? ______ What is the length of each step? _______ cm.
Part 3 — Doing the measurements
Before you start, here are a couple of tips for making measurements of a building.
- When you measure the building, always keep the cross-staff level with the ground when you are measuring the height of the building, being careful not to tilt it up or down.
- Take a series of 5 repeated measurements, varying the distance you are from the building.
- Then take the average of these as your best value, as well as stating how far above and below this average your individual data points varied. This is an estimate of the uncertainty in your measurements. For example: Average height was 30 ±0.3 meters. (Conversion of units from centimeters to meters involved.)
You will need to sight an object along the stick and slide the crosspiece back and forth, or change your own distance from the object, until the apparent or angular size of the crosspiece, or some portion of it, matches the angular size of the distant object. Find a building that can be far enough from you to view its total height.
Note that the angle from the top to the bottom of the crosspiece is the same as the angular size of the building, as shown in Figure 4. If D’ is the “working” length of the cross staff (in cm on the meter stick) and d’ the width shown on the crosspiece (in cm), then the angular size of the crosspiece is a’ = 57.3° (d’ / D’). But this value of a’ is also the angular size of the building, a. Thus, if you measure the distance D to the building, you can derive its height d. Logic:.
Since a’=a and the angles in degrees cancel, we get: d’/D’ = d/D. You get d’/D’ from your cross staff and you will count your steps and use your answer in question 3 to find D. Then, the height of the building is d = D×d’/D’.
Figure 4. Identifying the various factors involved in finding the height of a building.
3. Measure the distances to the building for each of the 5 measurements and complete the data in Table 1 to calculate its height, d. (Note that since d’ and D’ use the same units, those units cancel.)
Table 1. Measurements for finding the height of a building
| Measurement | Number of steps | Step size (cm) | Distance D (cm) | d’ (cm) | D’ (cm) | d’/D’ | d (cm) |
|---|---|---|---|---|---|---|---|
| 1 | |||||||
| 2 | |||||||
| 3 | |||||||
| 4 | |||||||
| 5 | |||||||
| Averages | |||||||
| Uncertainty ± | |||||||
4. Convert the height d in cm to height d in meters, expressing your uncertainty as given in the example above. Show your work here.
5. Briefly describe the building you selected. Were there any features that might have contributed to your determining its height? Examples might be structures or plants around its base and structures at its top such as additions for heating, cooling, pipes, or towers. Estimate how any of those features might have affected the uncertainty in the height of the building?
Part 4 — Summary
6. Describe a way you could use what you learned in this exercise to measure the size of another object. What kind of object would you choose? Why might it be important to know its actual size?
Figure 5. The pattern for the cross staff part of this activity. The pattern should be enlarged accordingly in order to fit the meter stick through its bottom slot.