Clinometer: Finding the height of objects using Isosceles Right Triangles

A clinometer, also called a declinometer or an inclinometer, is an instrument that measures vertical slope, usually the angle between the ground or the observer and a tall object. A simple, or fixed angle, clinometer requires plenty of room to walk back and forth when measuring objects frequently used in astronomy, surveying, engineering, and forestry.

Essentially, if you look through the sight of the clinometer and see the top of the object when the leg is horizontal to the ground and the other leg is vertical, your distance to the object's base is equal to the height of the object.

True story: This week by coincidence, an arborist working for the City of Boston was surveying a tree in front of our building in the South End for pruning. He was using a very expensive clinometer with lasers. I bet him that I could find the height of our tree within five feet of his precision reading. The Mathlete clinometer costs less than $1. I measured the tree at 77 feet and the arborist measured it and showed me the digital readout: 77 feet exactly. Then instead of collecting on the bet, I gave him one of my clinometers. Winning with math made all the difference.

1. Fold a piece of paper into a triangle. Fold the bottom right corner over to touch the left side of the paper, lining up the sides exactly to form a triangle. If you are using an ordinary rectangular sheet of paper, there will probably be an unfolded "extra" section above this triangle. Cut or tear this section off. What you are left with is an isosceles right triangle, with one 90º angle and two 45º angles.

• Construction paper will make a more durable clinometer, but you can use any sheet of paper. You may want to tape or glue the triangle together to make it sturdier.

2. Tape a straight drinking straw to the triangle's longest side. Position a drinking straw along the longest edge of the triangle, or hypotenuse, so that one end extends slightly out from the paper. Make sure the straw isn't bent or crushed, and runs straight along the hypotenuse. Use tape or glue to secure it to the paper. You will be looking through this straw when using the clinometer.

Sight the top of a tall object through the straw. Hold the longer end of the straw next to your eye and point it at the top of a tall object you want to measure, such as a tree. If you have to tilt the triangle in order to see the top of the object you're aiming for, you are too close or too far. 

Move forward or backward until base of the triangle is horizontal with the ground and the height of the triangle is perpendicular with the ground. In order to measure the tree, you need to find a spot to stand where you can hold the triangle completely flat and still see the top of the object through the straw. You can tell when the triangle is flat, because the weight will pull the string down exactly in line with one of the triangle's short sides.

• When this happens, it means the angle of elevation between your eye and the top of the object is 45 degrees.
• If you crouch or stand on an object to find a better position, you'll need to measure your height at eye level while in that position, instead of when standing normally as described in a later step.

3.  Use a tape measure to find the distance between this position and the base of the tall object. 

Just like the triangle you're holding, the giant triangle formed by you, the base of the tall object, and the top of the object has two 45º angles and one 90º angle. The two shorter sides of a 45-45-90 triangle are always the same length. Measure the distance between the position you were standing in at the end of the last step, and the base of the tall object you are measuring. The result is almost the height of the tall object, but there's one more step to get your final answer.

• If you don't have a tape measure, walk normally towards the tall object and count how many steps it takes to get there. Later, when you have a ruler, measure the length of one step and multiply by the number of steps you took to find the total distance (and therefore the height of the object).

MEASURING THE HEIGHT OF OBJECTS WITH A CLINOMETER

1. Find tall objects around your house or travels such as trees, telephone poles, tether ball poles, your house, neighboring houses, buildings downtown, even your room.

2. Follow the instructions on the reverse side and the activities we did in class to use your clinometer to measure the height of these objects.

3. Record your results below.

Name the Object Record the Height in Feet

Mr. Kramer's Building    65

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