Geometry

Volume and Surface Area: The Formulas You Actually Need

By Math Solving Space · · 2 min read

On this page
  1. Volume formulas
  2. Worked example 1: cylinder
  3. Worked example 2: cone
  4. Worked example 3: sphere
  5. Surface area formulas
  6. Worked example 4: surface area of a cylinder
  7. Worked example 5: surface area of a cone
  8. Common mistakes
  9. Practice questions
  10. Frequently asked questions

Volume is the space inside a 3D shape, measured in cubic units (cm³). Surface area is the total area of its outside, in square units (cm²). For prisms and cylinders, volume is base area × height. Cones and pyramids hold exactly one third of the matching prism.

Volume formulas

Solid Volume
Cube s3s^3
Cuboid lwhlwh
Cylinder πr2h\pi r^2 h
Cone 13πr2h\tfrac{1}{3}\pi r^2 h
Sphere 43πr3\tfrac{4}{3}\pi r^3
Square pyramid 13s2h\tfrac{1}{3}s^2 h

Worked example 1: cylinder

Radius 3 cm, height 5 cm:

V=πr2h=π×9×5=45π≈141.37 cm3V = \pi r^2 h = \pi \times 9 \times 5 = 45\pi \approx 141.37 \text{ cm}^3

Worked example 2: cone

Radius 4 cm, height 9 cm:

V=13π×16×9=48π≈150.80 cm3V = \tfrac{1}{3}\pi \times 16 \times 9 = 48\pi \approx 150.80 \text{ cm}^3

Worked example 3: sphere

Radius 6 m:

V=43π×216=288π≈904.78 m3V = \tfrac{4}{3}\pi \times 216 = 288\pi \approx 904.78 \text{ m}^3

Surface area formulas

Solid Surface area
Cube 6s26s^2
Cuboid 2(lw+lh+wh)2(lw + lh + wh)
Cylinder 2πr2+2πrh2\pi r^2 + 2\pi rh
Cone πr2+πrl\pi r^2 + \pi r l, where l=r2+h2l = \sqrt{r^2 + h^2}
Sphere 4πr24\pi r^2

Worked example 4: surface area of a cylinder

The cylinder has two circular ends and a curved side that unrolls into a rectangle of width 2πr2\pi r and height hh:

SA=2π(3)2+2π(3)(5)=18π+30π=48π≈150.80 cm2SA = 2\pi(3)^2 + 2\pi(3)(5) = 18\pi + 30\pi = 48\pi \approx 150.80 \text{ cm}^2

Worked example 5: surface area of a cone

Radius 3, height 4. First the slant height: l=9+16=5l = \sqrt{9 + 16} = 5.

SA=π(3)2+π(3)(5)=9π+15π=24π≈75.40SA = \pi(3)^2 + \pi(3)(5) = 9\pi + 15\pi = 24\pi \approx 75.40

Common mistakes

  • Using the vertical height hh instead of the slant height ll for a cone's curved surface.
  • Using the diameter instead of the radius.
  • Mixing up square and cubic units.

Practice questions

  1. Volume of a cube with side 4 cm.
  2. Volume of a hemisphere with radius 3 (exact).
  3. Surface area of a sphere with radius 2 (exact).

Answers: 1) 64 cm³ 2) 18π18\pi 3) 16π16\pi

The volume calculator and surface area calculator cover all these solids and keep π\pi exact. For flat shapes, see how to find the area of common shapes.

Frequently asked questions

Why is the sphere formula 4/3 πr³?

It can be derived with calculus, or by comparing a hemisphere with a cylinder minus a cone (Cavalieri's principle). The result is that a sphere fills two thirds of the cylinder around it.

What is the slant height?

On a cone or pyramid, it is the distance from the top down the sloping side to the edge of the base.

Do I need to include the base in surface area?

For a closed solid, yes. For an open container like a cup, leave out the missing face.

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