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Home›Geometry and drawings

Area and volume of solids

A cereal box, a tin can, an ice cream cone and a ball each have a formula for how much card covers them and how much they hold. Unfold each solid in 3D, change its size with sliders and watch the numbers follow.

Progress
  1. 1Nets and surface areaA cube with 4 cm edges is wrapped by exactly 96 cm² of card, and you can see all of it at once when the cube is opened flat.
  2. 2Volume of prismsA fish tank 50 cm long, 30 cm wide and 40 cm deep holds 60,000 cm³ of water, which is exactly 60 litres.
  3. 3CylindersPeel the label off a tin can and it lies flat as a rectangle whose width is the can's circumference, 2πr.
  4. 4Pyramids and conesA cone of water poured into a cylinder with the same base and height fills it exactly one third of the way.
  5. 5SpheresArchimedes asked for a sphere inside a cylinder to be carved on his tomb, because the sphere has exactly two thirds of the cylinder's volume.
  6. 6Faces, edges and vertices: Euler's formulaCount the faces, subtract the edges and add the vertices of any ordinary polyhedron and the answer is always 2.
  7. 7Scaling: double the lengthsMake every length of a box twice as long and it needs 4 times the card but holds 8 times as much.
Sources for this course (8)
  • (checked 2026-10-17; reordered 2026-10-19: curriculum and textbooks first, then reference works and units, history last)
  • Curriculum: 교육부 고시 제2022-33호 [별책 8] 수학과 교육과정 (ncic.re.kr/inv/org/download.do?year=2022&seq=10003559&orgType=ogi4, PDF read 2026-10-19: 초등학교 5~6학년군 [6수03-03]~[6수03-08] 직육면체·정육면체·각기둥·각뿔·원기둥·원뿔·구, 원기둥의 전개도, "각뿔과 원뿔의 전개도는 다루지 않는다", 원주율의 근삿값 3.14, [6수03-17]~[6수03-19] 직육면체와 정육면체의 겉넓이, 1cm³·1m³, 부피; 3~4학년군 [4수03-17] 들이 1L·1mL; 중학교 1~3학년군 [9수03-06] 부채꼴, [9수03-07] 다면체와 회전체, [9수03-08] 입체도형의 겉넓이와 부피, 도형의 닮음·닮음비; 오일러 공식은 성취기준에 이름으로 없음)
  • Textbook: OpenStax Prealgebra 2e, 9.6 Solve Geometry Applications: Volume and Surface Area (openstax.org/books/prealgebra-2e/pages/9-6-solve-geometry-applications-volume-and-surface-area)
  • Reference: Wolfram MathWorld, Polyhedral Formula (mathworld.wolfram.com/PolyhedralFormula.html)
  • AMS Feature Column, Euler's Polyhedral Formula (ams.org/publicoutreach/feature-column/fcarc-eulers-formula)
  • BIPM SI Brochure and NIST SI Units - Volume (litre and cubic metre)
  • History block only: MacTutor, Archimedes biography (On the Sphere and Cylinder, sphere = 2/3 of the circumscribed cylinder, the figure on his tomb, Cicero's search; mathshistory.st-andrews.ac.uk/Biographies/Archimedes/)
  • MacTutor, A history of Topology (Euler's 1750 letter to Goldbach with v − e + f = 2; mathshistory.st-andrews.ac.uk/HistTopics/Topology_in_mathematics)

Checked: 2026-10-17