Height Sight

1. Print out the protractor. Then, using your scissors, cut out the printed protractor, cutting carefully along the straight line on the top edge.
2. Tape the protractor to the 3 x 5 index card so the protractor’s straight side matches up with the card’s long side. Tape carefully around the curve of the protractor so it stays in place on the card.
3. Use the hole punch to punch a hole through the circle on the protractor.
4. Push one end of the string through the hole, and then through the washer. Tie the ends of the string together, making a loop on which the washer can freely slide.
5. Roll a sheet of 8.5 x 11 (A4) paper into a cylinder that’s 8.5 inches (22 cm) long, and about 1 inch (2.5 cm) in diameter. Tape the seam so the rolled-up paper stays in place.
6. Tape the paper cylinder to the card along the straight edge of the protractor. One end of the cylinder should line up with the edge of the card, as shown below.
   

Look through the free end of the tube, opposite the protractor, or sight along the tube’s top edge.

Focus on something at eye level. Ask your partner to read the angle where the string crosses the protractor. If your inclinometer is level, the string should cross the protractor at about 0 degrees.

If you’re trying this without a partner, look through the tube and pinch the string against the card to hold it in place. Then take the tube away from your eye and read the angle on the protractor. (Be careful not to move the string and change the angle!) Try this a few times, until you get the hang of it.

Now look through the tube at the top of something tall. If you’re indoors, look at something near the ceiling and find the angle where the string crosses the protractor. What do you think will happen if you change the distance between where you’re standing and the tall object you’re sighting? Make a guess, then take a few steps toward or away from the object and check your inclinometer. What happened to the angle?

Try this:

• Use your inclinometer to measure the top of something tall, but not too far away. Determine the angle on your inclinometer and write it down. This is the angular height of the object.
• Use your measuring tape to find out how far the base of that object is (in centimeters) from where you’re standing. Write down this distance.
• Have a partner measure how far it is from the floor to your eye level. Write down this distance.

You now have all the data you need to find the height of the object. You could use trigonometry to find the answer, but we’ve provided a handy Height Calculator Grid that will also do the job.

To begin, use a ruler to draw a line from the angle you measured through the origin and across the grid (see example in photo below).

Find the distance to your object along the horizontal (x) axis. Now note the scale of the grid. Draw a vertical line through that distance across the grid, and then find the place where the two lines intersect. Read the height along the vertical (y) axis. Notice the scale of the grid as you determine this height.

Notice that the x axis is labeled “eye level” because your measurement was taken from that height. To account for that measurement, add the height from the grid to the distance from your eye to the ground. This is the height of your object.

When you use your inclinometer to look at something above your head, the inclinometer tilts. The string crossing the protractor marks the angle of that tilt, which depends both on the height of the object and your distance to it. When you measure the angular height of something, your distance matters. Measure the angular height of the same tall object from two different distances, and you’ll get a larger angle when you’re closer, and a smaller angle when you’re farther away.

Calculating the height of an object that you cannot directly measure requires knowing both its angular height and your distance to the object. The Height Calculator Grid helps you determine the height of the object indirectly using similar triangles.

The triangle you drew on the grid is similar to the triangle formed in the real world because, for both triangles, all three angles are the same: The first common angle is the one you measured with your inclinometer, which is the same as the angle you drew on your grid. The second common angle is the 90-degree angle formed by the intersection of the vertical and horizontal lines (in this case, the vertical line of the object and the horizontal line marking the measurer’s eye level). The third common angle is the same by definition: the sum of the interior angles of any triangle equals 180 degrees.

That means the triangle on the grid is a scaled-down version of the triangle in the real world. By drawing a triangle that’s exactly the same shape as the triangle in the real world, you can measure something indirectly that you can't easily measure with a ruler.

In the Northern Hemisphere, you can also use your inclinometer to determine your latitude, your angular distance from the equator.

On a clear night, find the North Star (also known as Polaris). The easiest way to find the North Star is to find the Big Dipper. An imaginary arrow drawn through the two stars that form the end of the Big Dipper's bowl will point to the North Star, which is at the end of the Little Dipper’s handle. The North Star is always located between the Big Dipper and the constellation Cassiopeia. In the Northern Hemisphere, these constellations never set. Like the North Star, they are always in the sky on a clear night.

Sight on the North Star with your inclinometer. The angular height of the North Star, in degrees, is your latitude. The North Pole is at 90 degrees north latitude. Someone standing at the North Pole would have to look straight up to see the North Star.

You can also use your inclinometer to measure the altitude of the sun. The technique is different, because you must never look at the sun! Instead, hold the inclinometer at about waist level, and aim the tube at the sun. Make sure the string and washer hang freely. Hold your other hand below the tube, and adjust the position of the cylinder until the sun shines right through the tube and onto your hand. When the inclinometer is in this position, its angle is that of the sun’s altitude. Check the sun’s altitude at noon (or another hour) at different times of the year, or over the course of a day, to see how it changes with time.

This Science Snack is part of a collection that showcases female mathematicians and math educators whose work aids or expands our understanding of the phenomena explored in each Snack.

Source: Wikimedia Commons

Katherine Johnson (pictured above) was a mathematician and scientist who helped calculate our way to the stars. Johnson referred to her first department at NASA as the “computers who wore skirts,” a group of women doing the math for space exploration. In 1962, astronaut John Glenn told NASA that he would only go on his mission to orbit the Earth if Johnson confirmed the calculations from NASA’s computers. She provided the crucial orbit trajectories and went on to calculate them for 26 other missions including those to the moon and to Mars. With the Science Snack Height Sight, you can create a simple tool called an inclinometer that can be used to measure the height of distant objects such as the Sun.