Solving Building Height with Protractor and a Straw

The height of a building is determined with a drinking straw, a protractor, measuring tape and the right triangle.

Set straw at a 45 degree angle from the flat ground.

Looking through the straw, it is moved closer (or farther away) to the building so the top edge of the building is visible.

Looking through the other end of the straw, a coin is placed on the ground as a marker.

The coin marks the location the 45 degree angle touches the ground. The building meets the ground at the 90 degree angle. A right triangle has been created with the 90 degree angle, the 45 degree angle, and the top of the building. Measuring one side of the triangle, the height of the building is known.

According to geometry and trigonometry, the height of the building is also the same as the length between the base of the building and the coin. With the distance between the coin and the base of the building measured, the height of the building is known. When two sides of a right triangle are known, the third side is also known by using the Pythagorean theorem (a squared + b squared = c squared).

If the building base to coin is 16 feet, the height is 16 feet. The formula to determine top of building to coin is:

16 squared (16 * 16) height + 16 squared (16 * 16) base to coin =
512 (c squared) distance from coin to top

The square root of 512 feet is approximately 22.6 feet.

Trigonometry is the branch of Mathematics dealing with triangles.

The three basic functions are SINE (sin), COSINE (cos), and TANGENT (tan). The building height is a. The coin to building base is b. The building top to coin is c (hypotenuse).

a = building height = opposite
b = coin to building base = adjacent
c = building top to coin = hypotenuse

sin = a/c (opposite / hypotenuse)
cos = b/c (adjacent / hypotenuse)
tan = a/b (opposite / adjacent)
Cosecrant (cosec) = c/a
Secant (sec) = c/b
Cotangent (cot) = b/a

The building height can also be determined with the straw angle, distance from coin to base and Tangent function. The angle (45 degrees) and distance from the building (16 feet) are known.

tan 45 degrees = a/b = height/16 feet

The value of the Tangent function at 45 degrees is 1 (or 1/1). The trigonmetric function values are listed below.

height = 16 feet * 1

Another example..

In the drawing, a 30 degree angle and base to coin distance of 200 ft is provided. To determine the height of the red building, we determine the tangent decimal value for 30 degrees by checking the table.

The tangent decimal value for 30 degrees is 0.577. To solve for height, the tangent formula is utilized.

tan = a/b (opposite / adjacent)

The equation is updated with known values (tangent decimal value and adjacent value):

tan30 = a/200

The tangent decimal value for 30 degrees. tan30 = 0.577. The updated equation:

0.577 = a/200

Both sides are multiplied by 200 to solve for "a".

200*0.577 = a

115.4 = a

The building height in the drawing is approximately 115.4 ft.

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