NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
← Back to practice

Spatial problem-solving algorithms

Wide tables scroll horizontally. For detailed working, you may prefer a larger screen.

Required skills (click to reveal)

A designer is creating a three-dimensional scale model of a modular exhibition stand in Brisbane using digital modelling software. The stand consists of a central rectangular prism base measuring 3 m in length, 2 m in width, and 1.5 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 3 m (matching the length of the base) and a height of 2 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?

Press Start when ready. Time includes time spent in another tab. Submit your answer before opening a help panel for a timed rank.

00:00
0 wordsOnly your numerical final answer or selected MCQ option is checked. Scratch working is not graded.

Report a problem or suggest an improvement

My notes — optional, not graded 0 words

Notes and scratch working stay in this browser and are linked to this question version. They are not graded or included in feedback. On shared devices, other people using this browser may see them.

For this question only. Your full notebook is on the dashboard.

Notes save in this browser. Download a copy to keep them elsewhere.

Report a problem or suggest an improvement

Your observations help improve this question. You can report an issue or recommend a better explanation, visual or experience.

0 words

The report includes your message and question/version details. Saved profile details, answers, notes and attempt history are not automatically attached. Leave out names and contact details, and check screenshots for personal information and embedded metadata.

Attachments may contain personal information. Review them before sharing. Logs collect only event codes and timestamps after you enable them. Download the ZIP to keep a copy, then attach it to an email if you choose.

Formula and explanation (click to reveal)

Opening this before your first submission makes the attempt practice only.

Formula

Explanation

Worked solution and answer (click to reveal)

Opening this before your first submission makes the attempt practice only.

Worked solution

  1. The exhibition stand has a rectangular prism base measuring 3 m (length), 2 m (width), and 1.5 m (height). The triangular roof has a base of 3 m and a height of 2 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 length, width, and height of the 3D space from one corner to the opposite corner.
  3. The beam starts at a bottom corner of the base and ends at the apex of the roof, which is vertically above the centre of the base. The horizontal distance from the starting corner to the point directly below the apex is the diagonal of the base rectangle.
  4. Calculate the horizontal diagonal of the base: d_base = sqrt(3^2 + 2^2) = sqrt(9 + 4) = sqrt(13) m.
  5. The vertical rise from the base to the apex is 1.5 m (height of base) plus 2 m (height of roof), so total vertical height = 1.5 + 2 = 3.5 m.
  6. Now apply 3D Pythagoras: d = sqrt(d_base^2 + h_total^2) = sqrt((sqrt(13))^2 + 3.5^2) = sqrt(13 + 12.25) = sqrt(25.25) m.
  7. Simplify sqrt(25.25). Note that 25.25 = 101/4, so sqrt(101/4) = sqrt(101)/2.
  8. Calculate sqrt(101) ≈ 10.0498756, so d ≈ 10.0498756 / 2 ≈ 5.0249378 m.
  9. Check the calculation: 13 + 12.25 = 25.25, and sqrt(25.25) = sqrt(101)/2. This is exact.
  10. The beam length is exactly sqrt(101)/2 metres.
  11. The diagonal beam runs from a bottom corner to the opposite top apex, and the 3D path is correctly modelled using the full height and base diagonal.
  12. The exact length of the diagonal beam is sqrt(101)/2 metres.

Reference answer: sqrt(101)/2

Alternative methods (click to reveal)

Shortcuts are shown only when applicable. Check the conditions before using one.

Earlier attempts for this question

#Date / timeYour answerResultSecondsHelp revealedRank
History stays in this browser when storage is available. Export it to your shared Drive folder and import it on another computer.
Worksheet details

Naming pattern: Topic_Grade_Complexity_YYYYMMDD_HH24MISS.html

Validation compares your answer with the generated reference. The model's reference can be wrong. Time ranks use an estimated student solving time, not a measured benchmark or a comparison with other students. An incorrect answer receives no speed rank.

NavGanitMate privacy & saved work

Saved in this browser. Your profile preferences, answers, practice history and notes are stored here when browser storage is available. They are not automatically uploaded to our servers. People sharing this browser may see saved work; changing the learner name does not create a separate private account. A nickname is enough.

Keep copies of important work. Clearing browser data, private browsing or changing browser or device can mean losing saved work. Download notes and export history separately. Downloads are copies of the selected content, not a complete account backup; not every format can be imported to restore your work. Keep these files somewhere private.

Visiting the website. Cloudflare helps deliver and protect this site and processes connection information such as IP addresses and requested pages. Server error logs may also contain request details. Security features may use cookies. This is separate from your browser-saved learning records. Cloudflare privacy information.

A parent-connection reminder uses a local list of up to five completed question IDs for this browser profile and a tab-session dismissal marker. Reminder activity is not uploaded and practice remains available.

Sharing feedback is your choice. A report includes your message and question/version details. Your saved profile, answers and notes are not automatically attached. Files you select can reveal personal details or contain embedded metadata: check and redact them before sharing. Diagnostic event codes and timestamps are included only after you enable logging. Sending email shares your message and attachments with the recipient and email providers. A download alone sends nothing to us. Website feedback delivery may be unavailable; only a confirmed receipt indicates it was received.