ACT — mathMedium1 min read

Volume and Surface Area of 3D Shapes

Calculate volume and surface area of prisms, cylinders, cones, and spheres for the ACT.

Tutor AI Team
Tutor AI Team

The ACT tests your ability to calculate volumes and surface areas of common 3D shapes. Formulas may or may not be provided.

Key Formulas

Shape Volume Surface Area
Rectangular prism lwhlwh 2(lw+lh+wh)2(lw + lh + wh)
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 faces

Key Relationships

  • Cone = 13\frac{1}{3} of cylinder (same base, same height).
  • Doubling radius: area × 4, volume × 8.

Worked Example: Example 1

Problem

Cylinder: r=4r = 4, h=10h = 10. V=π(16)(10)=160πV = \pi(16)(10) = 160\pi. SA=32π+80π=112πSA = 32\pi + 80\pi = 112\pi.

Solution

Worked Example: Example 2

Problem

Sphere: r=6r = 6. V=43π(216)=288πV = \frac{4}{3}\pi(216) = 288\pi.

Solution

Practice Problems

    1. A cone has radius 3 and height 8. Find its volume.
    1. A cylinder and cone have the same radius and height. The cylinder's volume is 600π600\pi. Find the cone's volume.
    1. Find the surface area of a sphere with diameter 10.

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Key Takeaways

  • ✓

    Memorize the volume and surface area formulas.

  • ✓

    Cones and pyramids: 13\frac{1}{3} of the prism/cylinder equivalent.

  • ✓

    Watch radius vs diameter.

Tags:
#act#math#geometry#volume#surface-area

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