IB — mathsStandard1 min read

3D Shapes: Volume and Surface Area

Calculate volumes and surface areas for IB Maths. Apply to prisms, pyramids, cylinders, cones, and spheres.

Tutor AI Team
Tutor AI Team

3D geometry calculations are tested in both IB Math AI and AA. Formulas are in the data booklet.

Key Formulas

Shape Volume Surface Area
Prism Cross-section × length Sum of all faces
Cylinder πr2h\pi r^2 h 2πr2+2πrh2\pi r^2 + 2\pi rh
Cone 13πr2h\frac{1}{3}\pi r^2 h πrl+πr2\pi r l + \pi r^2
Sphere 43πr3\frac{4}{3}\pi r^3 4πr24\pi r^2
Pyramid 13Bh\frac{1}{3}Bh Base + lateral

Worked Example: Example 1

Problem

Hemisphere radius 6: V=23π(216)=144πV = \frac{2}{3}\pi(216) = 144\pi. SA=2π(36)+π(36)=108πSA = 2\pi(36) + \pi(36) = 108\pi.

Solution

Worked Example: Example 2

Problem

Cone: r=5r = 5, slant height l=13l = 13. Height: h=169−25=12h = \sqrt{169 - 25} = 12. V=13π(25)(12)=100πV = \frac{1}{3}\pi(25)(12) = 100\pi.

Solution

Practice Problems

    1. A sphere has surface area 100π100\pi. Find its volume.
    1. A cylinder and cone have same base and height. Cylinder volume is 270π270\pi. Find cone volume.

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Key Takeaways

  • ✓

    Formulas in data booklet — know when to use each.

  • ✓

    Cone/pyramid: 13\frac{1}{3} of prism/cylinder.

  • ✓

    Hemisphere: half sphere + circular base for SA.

Tags:
#ib#maths#geometry#volume#surface-area

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