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

Composite-object surface area and volume

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

Required skills (click to reveal)

A craft designer in Sydney creates a decorative solid by attaching a right circular cylinder of radius 24 cm and height 45 cm to the top of a rectangular prism with base dimensions 48 cm by 48 cm and height 24 cm. The cylinder is placed so that its base is perfectly aligned with the top face of the prism, and the interface between them is not exposed.

Calculate the exact total surface area of the composite solid in square centimetres, excluding the base of the prism (which rests on the floor) and the interface between the prism and cylinder. Express your answer as a single exact expression in terms of π.

Circle measurementsradius = 24 cmr = 24 cm
Measurements from the question · not to scale.
Concept illustrations

These are separate worked examples of related concepts, not a diagram of this question. Their values and dimensions do not describe your question.

Subtract a rectangular corner cut-out. The outer rectangle is 10 cm by 6 cm. Removing a 2 cm by 3 cm top-right corner leaves 60 − 6 = 54 cm². All dimensions are supplied; drawing is to one consistent scale.10 cm6 cm2 cm3 cm54 cm²
The outer rectangle is 10 cm by 6 cm. Removing a 2 cm by 3 cm top-right corner leaves 60 − 6 = 54 cm². All dimensions are supplied; drawing is to one consistent scale.
Area and perimeter of a rectangle. 6 × 4 = 24 square units inside; 6 + 4 + 6 + 4 = 20 units around.6 units across4
6 × 4 = 24 square units inside; 6 + 4 + 6 + 4 = 20 units around.

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 composite solid consists of a rectangular prism and a right circular cylinder attached at their bases. The base of the prism rests on the floor, so its base area is not part of the external surface area. The interface between the prism and cylinder is not exposed, so that area is excluded.
  2. The surface area of the composite solid is the sum of: (1) the lateral surface area of the rectangular prism, (2) the lateral surface area of the cylinder, and (3) the top face of the prism (which is not covered by the cylinder). The base of the prism is excluded, and the interface between the two solids is not exposed.
  3. The rectangular prism has base dimensions 48 cm × 48 cm and height 24 cm. The lateral surface area of a rectangular prism is 2 × (length + width) × height. Substituting: 2 × (48 + 48) × 24 = 2 × 96 × 24.
  4. Calculate: 2 × 96 = 192; 192 × 24 = 4608 cm². So, the lateral surface area of the prism is 4608 cm².
  5. The top face of the prism is a square of side 48 cm. Its area is 48 × 48 = 2304 cm². This face is exposed and must be included.
  6. The cylinder has radius 24 cm and height 45 cm. The lateral surface area of a right circular cylinder is 2πrh. Substituting: 2 × π × 24 × 45.
  7. Calculate: 2 × 24 = 48; 48 × 45 = 2160. So, lateral surface area of cylinder is 2160π cm².
  8. The base of the cylinder is attached to the prism and not exposed, so it is not included. The top of the cylinder is exposed and must be included. The top is a circle of radius 24 cm, so area is πr² = π × 24² = 576π cm².
  9. Now sum all external surface areas: lateral prism (4608 cm²) + top face of prism (2304 cm²) + lateral cylinder (2160π cm²) + top of cylinder (576π cm²).
  10. Combine like terms: 4608 + 2304 = 6912 cm² (flat areas). For the circular areas: 2160π + 576π = 2736π cm².
  11. The total surface area is 6912 + 2736π cm². This is the exact expression in terms of π, excluding the base of the prism and the interface.
  12. Final answer: 6912 + 2736π cm². This matches the required format: a single exact expression in terms of π.

Reference answer: 6912 + 2736π cm²

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.