NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Spatial problem-solving algorithms

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A designer is creating a three-dimensional scale model of a modular exhibition stand in Adelaide using digital modelling software. The stand consists of a central rectangular prism base measuring 4 m in length, 3 m in width, and 2 m in height, with a triangular roof section attached to the top face.

The roof is formed by two identical right-angled triangular faces, each with a base of 4 m (matching the length of the base) and a height of 2.5 m from the base to the apex. The designer needs to determine the exact length of the diagonal support beam that runs from one bottom corner of the base to the opposite top corner of the roof apex, passing through the interior of the structure.

Using spatial reasoning and the three-dimensional version of Pythagoras' theorem, what is the exact length of this diagonal beam in metres?

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Worked solution

  1. The exhibition stand has a rectangular prism base measuring 4 m (length), 3 m (width), and 2 m (height). The triangular roof has a base of 4 m and a height of 2.5 m from the base to the apex. The diagonal beam runs from a bottom corner of the base to the opposite top corner of the roof apex, passing through the interior.
  2. To find the length of the diagonal beam, apply the 3D version of Pythagoras' theorem: d = sqrt(l^2 + w^2 + h^2), where l, w, and h are the three perpendicular dimensions of the rectangular prism.
  3. The beam extends from a bottom corner of the base to the opposite top corner of the roof apex. The base is 4 m long, 3 m wide, and 2 m high. The roof adds vertical height from the top of the base to the apex, which is 2.5 m above the base. Therefore, the total vertical height from the base to the apex is 2 m (base height) + 2.5 m (roof height) = 4.5 m.
  4. The horizontal distance from the starting bottom corner to the opposite base corner is the diagonal of the base rectangle. Use 2D Pythagoras: base_diagonal = sqrt(length^2 + width^2) = sqrt(4^2 + 3^2) = sqrt(16 + 9) = sqrt(25) = 5 m.
  5. The diagonal beam forms a right-angled triangle in 3D space, with the base diagonal (5 m) as one leg, and the total vertical height (4.5 m) as the other leg. The beam is the hypotenuse of this triangle.
  6. Apply 3D Pythagoras: d = sqrt(base_diagonal^2 + total_vertical_height^2) = sqrt(5^2 + 4.5^2) = sqrt(25 + 20.25) = sqrt(45.25).
  7. Simplify sqrt(45.25). Note that 45.25 = 181/4, so sqrt(45.25) = sqrt(181/4) = sqrt(181)/2.
  8. Calculate sqrt(181). Since 13^2 = 169 and 14^2 = 196, and 181 is between them, sqrt(181) ≈ 13.4536. Therefore, d ≈ 13.4536 / 2 = 6.7268 m.
  9. Check the calculation: 5^2 = 25, 4.5^2 = 20.25, sum = 45.25. sqrt(45.25) = sqrt(181/4) = sqrt(181)/2. This is exact.
  10. The beam length is exactly sqrt(181)/2 metres. This is the simplest exact form.
  11. The diagonal beam runs from a bottom corner of the base to the opposite top corner of the roof apex, passing through the interior. The calculation accounts for all three dimensions: length, width, and total vertical height (base + roof). The result is exact.
  12. The exact length of the diagonal beam is sqrt(181)/2 metres.

Reference answer: sqrt(181)/2

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