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Trigonometry in the Right Triangle · Ages 15–16

Trigonometry Word Problems: Ladders, Ramps, and Shadows

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For 10th-grade students. Read each problem, draw and label a right triangle, then solve for the missing measurement. Round lengths to the nearest tenth and angles to the nearest degree.

  1. A 12-foot ladder leans against a vertical wall, making a 65-degree angle with the level ground. How high up the wall does the ladder reach?

  2. A wheelchair ramp rises 1.5 feet vertically over a horizontal distance of 18 feet. What angle does the ramp make with the ground?

  3. A vertical telephone pole casts a shadow that is 24 feet long on level ground when the angle of elevation of the sun is 52 degrees. How tall is the pole?

  4. A 20-foot ladder reaches a window sill that is 17 feet above the ground. What angle does the ladder make with the ground?

  5. A skateboard ramp is 10 feet long along its sloped surface and inclined at 22 degrees to the horizontal ground. How high is the top of the ramp off the ground?

  6. A monument stands 150 feet tall. When you measure its shadow, it is 85 feet long. What is the angle of elevation of the sun?

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Trigonometry Word Problems: Ladders, Ramps, and Shadows

Answer key

  1. A 12-foot ladder leans against a vertical wall, making a 65-degree angle with the level ground. How high up the wall does the ladder reach?

    10.9 feet. Set up sin(65 degrees) = height / 12, so height = 12 * sin(65 degrees) = 10.876 feet.

  2. A wheelchair ramp rises 1.5 feet vertically over a horizontal distance of 18 feet. What angle does the ramp make with the ground?

    5 degrees. Set up tan(theta) = 1.5 / 18 = 0.0833, so theta = arctan(0.0833) = 4.76 degrees.

  3. A vertical telephone pole casts a shadow that is 24 feet long on level ground when the angle of elevation of the sun is 52 degrees. How tall is the pole?

    30.7 feet. Set up tan(52 degrees) = height / 24, so height = 24 * tan(52 degrees) = 30.718 feet.

  4. A 20-foot ladder reaches a window sill that is 17 feet above the ground. What angle does the ladder make with the ground?

    58 degrees. Set up sin(theta) = 17 / 20 = 0.85, so theta = arcsin(0.85) = 58.21 degrees.

  5. A skateboard ramp is 10 feet long along its sloped surface and inclined at 22 degrees to the horizontal ground. How high is the top of the ramp off the ground?

    3.7 feet. Set up sin(22 degrees) = height / 10, so height = 10 * sin(22 degrees) = 3.746 feet.

  6. A monument stands 150 feet tall. When you measure its shadow, it is 85 feet long. What is the angle of elevation of the sun?

    60 degrees. Set up tan(theta) = 150 / 85 = 1.7647, so theta = arctan(1.7647) = 60.46 degrees.