Rectangular Box and Prism Formulas

Volume, surface area, the sum of the edge lengths and the space diagonal of a rectangular box, plus the volume and the lateral and total surface area of a prism.

8 formulas · Grades 5–11

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All 8 formulas on this page
  1. Volume of a rectangular box
  2. Surface area of a rectangular box
  3. Sum of edge lengths of a rectangular box
  4. Lateral surface area of a rectangular box
  5. Volume of a prism
  6. Lateral surface area of a prism
  7. Diagonal of a rectangular box
  8. Total surface area of a regular prism

Volume of a rectangular box

Grade 5
V=a⋅b⋅cV = a \cdot b \cdot c

Product of the three dimensions: length, width and height.

acb
a, b, c are the three dimensions of the box.
Legend:
  • a — Length (a)
  • b — Width (b)
  • c — Height (c)

Surface area of a rectangular box

Grade 5
S=2⋅(ab+bc+ac)S = 2 \cdot (ab + bc + ac)

Sum of the areas of all six faces of the box.

acb
The areas of all six faces are added together.
Legend:
  • a — Edge a
  • b — Edge b
  • c — Edge c

Sum of edge lengths of a rectangular box

Grade 5
L=4⋅(a+b+c)L = 4 \cdot (a + b + c)

Sum of the lengths of all 12 edges of a rectangular box.

acb
A rectangular box has 4 edges of each dimension.
Legend:
  • a — Edge a
  • b — Edge b
  • c — Edge c

Lateral surface area of a rectangular box

Grade 5
S=2c(a+b)S = 2c(a+b)

Lateral surface area of a rectangular box from its three dimensions.

acb
Lateral surface = the four vertical faces.
Legend:
  • a — Edge a
  • b — Edge b
  • c — Edge c

Volume of a prism

Grade 10
V=S⋅hV = S \cdot h

The volume of a prism equals the base area times the height.

hS
S is the base area, h is the height of the prism.
Legend:
  • S — Base area (S)
  • h — Height (h)

Lateral surface area of a prism

Grade 10
Sb=P⋅hS_b = P \cdot h

Lateral surface area of a right prism via the base perimeter and the height.

hS
Lateral surface = all the side faces of the prism.
Legend:
  • P — Perimeter (P)
  • h — Height (h)

Diagonal of a rectangular box

Grade 11
d=a2+b2+c2d = \sqrt{a^2+b^2+c^2}

Diagonal of a rectangular box from its three dimensions.

dacb
d is the diagonal connecting opposite vertices.
Legend:
  • a — Edge a
  • b — Edge b
  • c — Edge c

Total surface area of a regular prism

Grade 11
S=2S0+P0⋅hS = 2S_0 + P_0 \cdot h

Sum of the areas of the two bases and the lateral surface of a right prism.

hS
Total surface = lateral surface plus two bases.
Legend:
  • Sb — Base area (S)
  • Pb — Perimeter (P)
  • h — Height (h)

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