Cuboids, prisms and exposed faces
Multiply a uniform cross-section by length for volume, then count exposed faces for surface area.
Core and Extended.
Before you begin
- Find areas of rectangles and triangles.
- Interpret a solid's net.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is the area of a right triangle with perpendicular sides 3 and 4 cm?
What you will learn
- Find volume and surface area of cuboids and right prisms.
- Interpret a uniform cross-section and distinguish open from closed solids.
Stack the same cross-section
A prism has a uniform cross-section along its length. Its volume is cross-sectional area times perpendicular length. A right triangular prism with a 3–4–5 cm cross-section and length 10 cm has cross-sectional area 3 × 4/2 = 6 cm² and volume 60 cm³. The length between its matching ends is not a triangle side.
A cuboid is a rectangular prism: volume abc for three perpendicular edge lengths. If volume and cross-sectional area are known, divide to find the length. The uniform cross-section may be more complex than a polygon, including a sector-shaped end; find that end's area first. For a 90° sector of radius 6 cm, end area is 9π cm²; a 5 cm long prism then has volume 45π cm³. Its end perimeter is 3π + 12 cm, so closed surface area is 2(9π) + 5(3π + 12) = 33π + 60 cm². Use matching length units before multiplying.
Surface area needs each boundary face
A closed right prism has two matching ends plus rectangles along every cross-section edge. Its area is 2A + pL, where A is end area, p its perimeter and L the prism length. For the 3–4–5 prism above, this is 12 + (3 + 4 + 5) × 10 = 132 cm². A cuboid gives 2(ab + ac + bc).
For an open-top box, remove only the missing top face from the closed area. Do not remove the base or all side faces. A painted or wrapped solid may expose fewer surfaces than a closed net; draw a face list before calculating and distinguish a face area from its boundary length.
Pause and explain
A prism has cross-sectional area 12 cm² and length 7 cm. Find its volume.
Worked example
Find volume and closed surface area of a right triangular prism with 3–4–5 cm cross-section and length 10 cm.
Show the worked solution
- Cross-section area is 3 × 4/2 = 6 cm²; volume is 6 × 10 = 60 cm³.
- The three side rectangles total (3 + 4 + 5) × 10 = 120 cm².
- Add two triangular ends, 12 cm², giving surface area 132 cm².
Answer Volume 60 cm³; surface area 132 cm².
From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A cuboid has edges 2, 3 and 5 cm. Find volume and closed surface area.
Give me a hint
List its three pairs of equal faces.
Compare my reasoning
- Volume is 2 × 3 × 5 = 30 cm³.
- One of each face pair has area 6, 10 and 15 cm².
- Double their sum: surface area 62 cm².
30 cm³; 62 cm².
Look for these in your work
- Used three dimensions for volume.
- Included all six faces.
A prism has cross-sectional area 18 cm² and volume 216 cm³. Its end perimeter is 20 cm. Find its length and closed surface area.
Give me a hint
Recover length before the side areas.
Compare my reasoning
- Length is 216/18 = 12 cm.
- Side area is 20 × 12 = 240 cm².
- The two ends add 36 cm², giving 276 cm².
Length 12 cm; surface area 276 cm².
Look for these in your work
- Divided volume by cross-sectional area.
- Included both ends.
An open-top rectangular box has base 8 by 5 cm and height 3 cm. Find capacity and material area, ignoring thickness.
Give me a hint
The top is missing but the base remains.
Compare my reasoning
- Capacity is 8 × 5 × 3 = 120 cm³.
- The base is 40 cm²; side pairs are 2 × 24 and 2 × 15 cm².
- Material area is 40 + 48 + 30 = 118 cm².
Capacity 120 cm³; material area 118 cm².
Look for these in your work
- Included exactly the exposed five faces.
- Distinguished capacity from material area.
Common mistakes
- Using a triangle's perimeter as its area.
- Omitting one prism end.
- Including an absent lid in an open box's material area.
What can you explain now?
What must be uniform along a prism?
Compare with the explanation
Its cross-section.
Equal cross-sectional areas and shape at every position let volume equal area times length.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, sketch a right prism net and derive its surface area from two ends and the side strip.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs 21; Coordinate geometry 10; Geometry 15; Mensuration 13; Trigonometry 13; Transformations and vectors 11; Probability 7; Statistics 13. All 72 syllabus section entries now link to teaching and staged practice. Seven mixed checkpoints sample skills and suggest review lessons. Nine earlier overviews remain available. Full cumulative assessment and human teacher review remain pending. Written lesson practice, charts and constructions are self-checked, not automatically graded. Checkpoints do not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources