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Pythagorean 3D

Grade 8 · Geometry · Worksheet 3

  1. Sophia is helping her school's drama club build a set for an upcoming play. They are constructing a large rectangular storage box to hold props backstage. The interior dimensions of the box are 18 feet long, 15 feet wide, and 12 feet high. Sophia needs to place a long, straight decorative pole diagonally from the bottom front left corner of the box to the top back right corner. What is the exact length of the longest pole that can fit in a straight line inside the box? Answer: ______________
  2. Emma is building a model of a tower for her school project. The tower is shaped like a rectangular prism with a base that is 7 cm long and 5 cm wide, and a height of 11 cm. She wants to run a thin wire from the bottom front left corner of the tower to the top back right corner, going straight through the interior. What is the exact length of the wire she needs?
    Answer: ______________
  3. √(11² + 60² + 61²) = ? Answer: ______________
  4. A rectangular prism has dimensions 7 cm × 12 cm × 27 cm. Find the length of the space diagonal from one vertex to the opposite vertex.
    Answer: ______________
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Answer Key & Explanations

Pythagorean 3D · Grade 8 · Worksheet 3

  1. Sophia is helping her school's drama club build a set for an upcoming play. They are constructing a large rectangular storage box to hold props backstage. The interior dimensions of the box are 18 feet long, 15 feet wide, and 12 feet high. Sophia needs to place a long, straight decorative pole diagonally from the bottom front left corner of the box to the top back right corner. What is the exact length of the longest pole that can fit in a straight line inside the box? Answer: sqrt(693) feet or approximately 26.3 feet Solution: Find the diagonal of the base (floor). The base is a rectangle with length 18 ft and width 15 ft. Using the Pythagorean theorem: d_base^2 = 18^2 + 15^2 = 324 + 225 = 549.
    Full step-by-step solution

    Step 1: Find the diagonal of the base (floor). The base is a rectangle with length 18 ft and width 15 ft. Using the Pythagorean theorem: d_base^2 = 18^2 + 15^2 = 324 + 225 = 549. So d_base = sqrt(549). Step 2: Now, the space diagonal (the pole) is the hypotenuse of a right triangle where one leg is the base diagonal (sqrt(549)) and the other leg is the height (12 ft). Apply the Pythagorean theorem again: d_pole^2 = (sqrt(549))^2 + 12^2 = 549 + 144 = 693. Step 3: Take the square root: d_pole = sqrt(693). This is the exact length. Step 4: To approximate, note that 693 = 9 * 77, so sqrt(693) = 3 * sqrt(77). Since sqrt(77) is approximately 8.775, then d_pole is about 26.3 ft (rounded to the nearest tenth). The longest pole that can fit is sqrt(693) feet, or approximately 26.3 feet.

  2. Emma is building a model of a tower for her school project. The tower is shaped like a rectangular prism with a base that is 7 cm long and 5 cm wide, and a height of 11 cm. She wants to run a thin wire from the bottom front left corner of the tower to the top back right corner, going straight through the interior. What is the exact length of the wire she needs? Answer: sqrt(195) cm Solution: Find the diagonal of the base. The base is a rectangle with length 7 cm and width 5 cm. Using the Pythagorean theorem: base diagonal squared = 7^2 + 5^2 = 49 + 25 = 74, so base diagonal = sqrt(74) cm.
    Full step-by-step solution

    Step 1: Find the diagonal of the base. The base is a rectangle with length 7 cm and width 5 cm. Using the Pythagorean theorem: base diagonal squared = 7^2 + 5^2 = 49 + 25 = 74, so base diagonal = sqrt(74) cm. Step 2: Now consider a vertical right triangle where one leg is the base diagonal (sqrt(74) cm) and the other leg is the height (11 cm). The space diagonal (the wire) is the hypotenuse. Step 3: Apply the Pythagorean theorem again: space diagonal squared = (sqrt(74))^2 + 11^2 = 74 + 121 = 195. Step 4: Take the square root: space diagonal = sqrt(195) cm. The exact length of the wire Emma needs is sqrt(195) cm.

  3. √(11² + 60² + 61²) = ? Answer: 86 Solution: Square each dimension: 11² = 121, 60² = 3600, 61² = 3721 Add the squares: 121 + 3600 + 3721 = 7442 Take the square root: √7442 = 86 The answer is 86.
    Full step-by-step solution

    Step 1: Square each dimension: 11² = 121, 60² = 3600, 61² = 3721 Step 2: Add the squares: 121 + 3600 + 3721 = 7442 Step 3: Take the square root: √7442 = 86 The answer is 86.

  4. A rectangular prism has dimensions 7 cm × 12 cm × 27 cm. Find the length of the space diagonal from one vertex to the opposite vertex. Answer: sqrt(7² + 12² + 27²) = sqrt(49 + 144 + 729) = sqrt(922) ≈ 30.36 cm Solution: Find the diagonal of the base (7 cm × 12 cm). Using the Pythagorean theorem: d_base = sqrt(7² + 12²) = sqrt(49 + 144) = sqrt(193) cm. Now use this base diagonal with the height (27 cm) to find the space diagonal.
    Full step-by-step solution

    Step 1: Find the diagonal of the base (7 cm × 12 cm). Using the Pythagorean theorem: d_base = sqrt(7² + 12²) = sqrt(49 + 144) = sqrt(193) cm. Step 2: Now use this base diagonal with the height (27 cm) to find the space diagonal. The space diagonal forms a right triangle with the base diagonal and the height: d_space = sqrt((sqrt(193))² + 27²) = sqrt(193 + 729) = sqrt(922). Step 3: Simplify sqrt(922). Since 922 = 2 × 461 and 461 is prime, sqrt(922) cannot be simplified further. Approximate value: sqrt(922) ≈ 30.36 cm. The answer is sqrt(922) cm or approximately 30.36 cm.