A slant height is not measured at a 90-degree angle from the base. The most common occurrence of slant height is with the use of ladders. When a ladder is placed against a house, the distance from the ground to the top of the ladder is not known. However, the length of a ladder is known. The problem is solved by making a right triangle out of the wall, ladder and ground and taking some measurements.
If the Distance of the Base Is Known
Create a right triangle out of the slant height, regular height and base. The right angle is between the base and the regular height.
Square the slant height and the length of the base. For example, if the base is 3 feet and the slant height is 5 feet, then take 3^2 and 5^2 to yield 9 ft^2 and 25 ft^2, respectively.
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Subtract the base length squared from the slant height squared. In this example, evaluate 25 ft^2 minus 9 ft^2 to yield 16 ft^2.
Evaluate the square root of the result from Step 3. In this example, the square root of 16 ft^2 is 4 feet, which is the regular height.
If the Angle of the Slant Height Is Known
Create a right triangle out of the slant height, regular height and base. The right angle is between the base and the regular height. The angle of the slant height is between the base and the slant height.
Use the laws of trigonometry to create an equation for the regular height. In this example, the sine of the slant height angle is equal to the length of the regular height over the length of the slant height. In equation form, this yields sin(angle) = regular height/slant height.
Evaluate the equation from the previous step to yield the regular height. For example, if the slant height angle is 30 degrees and the slant height is 20 feet, then use the equation sin(30) = regular height / 20 feet. This yields 10 feet as the regular height.