Solid Geometry Formulas

Volumes and surface areas of the cube, rectangular box, prism, pyramid, cylinder, cone and sphere, along with diagonals, cross-sections and parts of a sphere — the main formulas of solid geometry in one place.

35 formulas · Grades 5–11

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All 35 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. Volume of a cube
  5. Surface area of a cube
  6. Lateral surface area of a rectangular box
  7. Sum of edge lengths of a cube
  8. Volume of a prism
  9. Volume of a pyramid
  10. Lateral surface area of a prism
  11. Volume of a cylinder
  12. Lateral surface area of a cylinder
  13. Volume of a cone
  14. Lateral surface area of a cone
  15. Volume of a sphere
  16. Surface area of a sphere
  17. Diagonal of a rectangular box
  18. Diagonal of a cube
  19. Total surface area of a regular prism
  20. Total surface area of a cylinder
  21. Area of the axial cross-section of a cylinder
  22. Diagonal of the axial cross-section of a cylinder
  23. Surface area of a sphere via diameter
  24. Area of a spherical cap's curved surface
  25. Volume of a spherical cap
  26. Volume of a spherical sector
  27. Lateral surface area of a pyramid
  28. Total surface area of a pyramid
  29. Total surface area of a cone
  30. Lateral surface area of a frustum of a cone
  31. Total surface area of a frustum of a cone
  32. Sum of edge lengths of a regular n-gonal pyramid
  33. Sum of edge lengths of a regular n-gonal prism
  34. Sum of edge lengths of a regular tetrahedron
  35. Sum of edge lengths of a regular octahedron

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

Volume of a cube

Grade 5
V=a3V = a^3

The cube of the edge length.

aaa
All edges of a cube are equal.
Legend:
  • a — Edge a

Surface area of a cube

Grade 5
S=6⋅a2S = 6 \cdot a^2

Sum of the areas of the six equal square faces.

aaa
A cube has 6 identical square faces.
Legend:
  • a — Edge a

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

Sum of edge lengths of a cube

Grade 5
L=12aL = 12a

Sum of the lengths of all 12 edges of a cube.

a
All 12 edges of the cube have the same length a — highlighted in green.
Legend:
  • a — Edge a

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)

Volume of a pyramid

Grade 10
V=13S⋅hV = \frac{1}{3} S \cdot h

The volume of a pyramid equals one third of the base area times the height.

hS
S is the base area, h is the height of the pyramid.
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)

Volume of a cylinder

Grade 11
V=πr2hV = \pi r^2 h

Volume of a cylinder with base radius r and height h.

rh
r is the base radius, h is the height of the cylinder.
Legend:
  • r — Radius (r)
  • h — Height (h)

Lateral surface area of a cylinder

Grade 11
S=2πrhS = 2 \pi r h

Lateral surface area of a right circular cylinder.

rh
The lateral surface unrolls into a rectangle.
Legend:
  • r — Radius (r)
  • h — Height (h)

Volume of a cone

Grade 11
V=13πr2hV = \frac{1}{3} \pi r^2 h

Volume of a right circular cone.

rh
r is the base radius, h is the height of the cone.
Legend:
  • r — Radius (r)
  • h — Height (h)

Lateral surface area of a cone

Grade 11
S=πrlS = \pi r l

Lateral surface area of a cone via radius r and slant height l.

rl
l is the slant height of the cone, r is the base radius.
Legend:
  • r — Radius (r)
  • l — Slant height (l)

Volume of a sphere

Grade 11
V=43πR3V = \frac{4}{3} \pi R^3

Volume of a sphere of radius R.

R
R is the radius of the ball, from the centre to the surface.
Legend:
  • r — Radius (R)

Surface area of a sphere

Grade 11
S=4πR2S = 4 \pi R^2

Surface area of a sphere of radius R.

R
The area of the entire outer surface of the ball.
Legend:
  • r — Radius (R)

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

Diagonal of a cube

Grade 11
d=a3d = a\sqrt{3}

Diagonal of a cube from its edge length.

daaa
The diagonal of a cube passes through its centre.
Legend:
  • a — Edge a

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)

Total surface area of a cylinder

Grade 11
S=2πr(r+h)S = 2\pi r(r+h)

Sum of the lateral surface area and the two bases of a cylinder.

rh
Total surface = lateral surface plus two circles.
Legend:
  • r — Radius (r)
  • h — Height (h)

Area of the axial cross-section of a cylinder

Grade 11
S=2rhS = 2rh

Area of the rectangle formed by a cylinder's axial cross-section.

drh
The axial cross-section is a rectangle through the cylinder's axis.
Legend:
  • r — Radius (r)
  • h — Height (h)

Diagonal of the axial cross-section of a cylinder

Grade 11
D=(2r)2+h2D = \sqrt{(2r)^2+h^2}

Diagonal of the rectangle of a cylinder's axial cross-section (via the Pythagorean theorem).

drh
d is the diagonal of the cylinder's axial cross-section.
Legend:
  • r — Radius (r)
  • h — Height (h)

Surface area of a sphere via diameter

Grade 11
S=πd2S = \pi d^2

Surface area of a sphere expressed through its diameter.

Legend:
  • d — Diameter (d)

Area of a spherical cap's curved surface

Grade 11
S=2πRhS = 2\pi R h

Area of the part of a sphere's surface cut off by a plane, of height h.

Legend:
  • R — Radius (R)
  • h — Height of the segment (h)

Volume of a spherical cap

Grade 11
V=πh2(R−h3)V = \pi h^2 \left(R - \frac{h}{3}\right)

Volume of the part of a ball cut off by a plane, where h is the cap's height.

Legend:
  • R — Radius (R)
  • h — Height of the segment (h)

Volume of a spherical sector

Grade 11
V=23πR2hV = \frac{2}{3}\pi R^2 h

Volume of the solid made of a spherical cap and a cone, where h is the height of the corresponding cap.

Legend:
  • R — Radius (R)
  • h — Height of the segment (h)

Lateral surface area of a pyramid

Grade 11
Slat=12P⋅lS_{lat} = \frac{1}{2} P \cdot l

Lateral surface of a regular pyramid via the base perimeter and the apothem.

lP
l is the apothem (the height of a lateral face), P is the base perimeter.
Legend:
  • P — Base perimeter (P)
  • l — Apothem (l)

Total surface area of a pyramid

Grade 11
Sfull=Slat+SbaseS_{full} = S_{lat} + S_{base}

Total surface of a pyramid — the lateral surface plus the base area.

SбокSосн
Total surface = lateral surface plus the base.
Legend:
  • Sbok — Lateral surface
  • Sosn — Base area

Total surface area of a cone

Grade 11
S=πrl+πr2=πr(l+r)S = \pi r l + \pi r^2 = \pi r (l + r)

Total surface of a cone — the lateral surface plus the base area.

rl
Total surface = lateral surface plus the circular base.
Legend:
  • r — Radius (r)
  • l — Slant height (l)

Lateral surface area of a frustum of a cone

Grade 11
Slat=π(R1+R2)lS_{lat} = \pi (R_1 + R_2) l

Lateral surface of a frustum of a cone from the base radii and the slant height.

R₁R₂l
R₁ and R₂ are the radii of the two bases, l is the slant height.
Legend:
  • R1 — Radius R1
  • R2 — Radius R2
  • l — Slant height (l)

Total surface area of a frustum of a cone

Grade 11
Sfull=π(R1+R2)l+πR12+πR22S_{full} = \pi (R_1 + R_2) l + \pi R_1^2 + \pi R_2^2

Total surface of a frustum of a cone — the lateral surface plus the areas of the two bases.

R₁R₂l
Total surface = lateral surface plus the two bases.
Legend:
  • R1 — Radius R1
  • R2 — Radius R2
  • l — Slant height (l)

Sum of edge lengths of a regular n-gonal pyramid

Grade 11
L=n(a+b)L = n(a + b)

Sum of the lengths of all edges of a regular pyramid: n base sides plus n lateral edges.

ab
a is the base side, b is the lateral edge (n of each).
Legend:
  • n — Number of base sides (n)
  • a — Base side (a)
  • b — Lateral edge (b)

Sum of edge lengths of a regular n-gonal prism

Grade 11
L=n(2a+h)L = n(2a + h)

Sum of the lengths of all edges of a regular prism: the edges of the two bases plus the lateral edges.

ah
a is the base side, h is the height (lateral edge) of the prism.
Legend:
  • n — Number of base sides (n)
  • a — Base side (a)
  • h — Height of the prism (h)

Sum of edge lengths of a regular tetrahedron

Grade 11
L=6aL = 6a

Sum of the lengths of all 6 edges of a regular tetrahedron.

a
A tetrahedron has 4 triangular faces, all 6 edges equal a.
Legend:
  • a — Edge a

Sum of edge lengths of a regular octahedron

Grade 11
L=12aL = 12a

Sum of the lengths of all 12 edges of a regular octahedron.

a
An octahedron has 8 triangular faces, all 12 edges equal a.
Legend:
  • a — Edge a

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