How to Estimate the Height of Objects With a Triangle

If you know the distance from the golf ball to the flag pole, and the angle from the golf ball to the top of the flag, you can find the height of the flag pole.

There are two ways to use the properties of a triangle to estimate the height of objects. Which method you use will depend on the information you have available. If you know the distance along the ground from the object to the next point of your triangle and the distance from that point to the top of the object, you can use the Pythagorean theorem -- a-squared + b-squared = c-squared -- to find the height. If you know one distance and one angle, such as the angle looking up from the ground, you can use sine, cosine or tangent depending on which side and angle you have.

Things You'll Need

• Pencil
• Paper
• Calculator
• Sine/cosine/tangent tables if this function is not on your calculator

Instructions

Finding Height with Pythagorean Theorem

1. Draw a picture of a triangle on a piece of paper. The bottom leg is the ground, the hypotenuse is the distance from the point on the ground to the top of the object, and the other leg is the object whose height you are finding.

2. Plug the numbers you have into the equation, where the ground distance is a and the distance from the point on the ground is c. For example, if your object casts a 10-foot shadow and the distance from the end of the shadow to the top of the object is 15 feet, your equation would be 10 squared + b-squared = 15 squared.

3. Simplify the equation by solving for the squares: 100 + b-squared = 225.

4. Subtract your a-squared from both sides to get b-squared by itself: b-squared = 125.

5. Find the square root of each side to solve for b: b = 11.18 (rounded to nearest hundredth). So the height of your object is approximately 11 feet.

Find the Height of Your Object Using Sine, Cosine or Tangent

1. Draw a picture of a triangle on a piece of paper. The bottom leg is the ground, the hypotenuse is the distance from the point on the ground to the top of the object, and the other leg is the object whose height you are finding.

2. Label the pieces of the triangle that you have. If you have the angle from the ground looking up and the hypotenuse length, you will use sine. If you have an angle from the ground looking up and the leg next to the angle (not the hypotenuse), you will use tangent. If you have the angle looking down from the top of the object and the leg opposite the angle, you will use tangent. And if you have the angle looking down from the top of the object and the hypotenuse, you will use cosine.

3. Plug the numbers into the equation you will be using. In the case of sine, angle = length of opposite leg divided by length of hypotenuse. With cosine, angle = length of adjacent leg divided by length of hypotenuse. With tangent, angle = length of opposite leg divided by length of adjacent leg.

4. Use your calculator or tables to find the sine, cosine or tangent of the angle, depending on which formula you are using. Put this number in place of the angle in the formula. On a scientific calculator, you would input the angle. For example, a 56-degree angle, then hit the sin, cos or tan key. You will get a decimal number; round it to four digits. Sine of a 56 degree angle would be .8290.

5. Solve for the height of the object. In the example using sine, with a 56-degree angle and a hypotenuse of 15 feet, your equation would be .8290 = x divided by 15. Multiply both sides by 15 to get x by itself: x = 12.435.

Tips & Warnings

• Show your work and draw pictures. It is better to have too much work than to miss something because you didn't write it down.

References

• The Institute for Environmental Learning: Estimation of Tree Height: Right Triangle Trigonometry
• University of Georgia: The Pythagorean Theorem: Stephanie J. Morris

Resources

• National Aeronautics and Space Administration: Sine-Cosine-Tangent