The surface area to volume ratio calculator effortlessly compute the relationship between an object’s external surface area and its internal volume in biology, engineering, and material science.
Cone
- Surface Area: S = πr(r + √(h² + r²))
- Volume: V = (1/3)πr²h
Pyramid
- Surface Area: S = (b²/2) + b * l (where b is the base length and l is the slant height)
- Volume: V = (1/3)b²h
Surface Area to Volume Ratio Calculator
| Shape | Dimensions | Surface Area | Volume | SA:V Ratio |
|---|---|---|---|---|
| Cube | side = 3 cm | 54 cm² | 27 cm³ | 2 cm^-1 |
| Sphere | radius = 2 cm | 50.27 cm² | 33.51 cm³ | 1.5 cm^-1 |
| Cylinder | radius = 1 cm, height = 4 cm | 31.42 cm² | 12.57 cm³ | 2.5 cm^-1 |
| Cuboid | length = 4 cm, width = 2 cm, height = 3 cm | 52 cm² | 24 cm³ | 2.17 cm^-1 |
| Rectangular Prism | length = 6 cm, width = 3 cm, height = 2 cm | 66 cm² | 36 cm³ | 1.83 cm^-1 |
| Cone | radius = 2 cm, height = 5 cm | 37.70 cm² | 13.33 cm³ | 2.83 cm^-1 |
| Pyramid | base side = 3 cm, height = 4 cm (square base) | 27 cm² | 12 cm³ | 2.25 cm^-1 |
| Ellipsoid | semi-major axis = 3 cm, semi-minor axis = 2 cm, semi-minor axis = 1.5 cm | ~37.69 cm² | ~28.27 cm³ | ~1.33 cm^-1 |
| Torus | major radius = 3 cm, minor radius = 1 cm | ~62.83 cm² | ~18.85 cm³ | ~3.33 cm^-1 |
| Triangular Prism | base side = 3 cm, height of prism = 5 cm | ~31.18 cm² | ~15.75 cm³ | ~1.98 cm^-1 |
Surface Area to Volume Ratio Formula
The formula for calculating the surface area to volume ratio is:
SA:V Ratio = Total Surface Area / Total VolumeThis ratio is typically expressed as a unit-less number or with units of length^-1 (such as mm^-1 or cm^-1).
For a cube with side length 2 cm:
- Surface Area = 6 × (2 cm)² = 24 cm²
- Volume = (2 cm)³ = 8 cm³
- SA:V Ratio = 24/8 = 3 cm^-1
Ellipsoid
- Surface Area: Approximation using S ≈ 4π((a^p b^p + a^p c^p + b^p * c^p)/3)^(1/p) (where p ≈ 1.6075)
- Volume: V = (4/3)πabc
Torus
- Surface Area: S = (2πR)(2πr)
- Volume: V = (2πR)(πr²)
Triangular Prism
- Surface Area: A = bh + (s₁ + s₂ + s₃)h (where b is the base length and s₁, s₂, s₃ are the lengths of the sides)
- Volume: V = (1/2)bhL
How to Calculate Surface Area to Volume Ratio?
- First, calculate the surface area (SA) of the object using the appropriate formula for its shape.
- Then, calculate the volume (V) of the object.
- Finally, divide the surface area by the volume: SA:V = Surface Area ÷ Volume
For a Cube (side length = a):
- Surface Area = 6a²
- Volume = a³
- SA:V = 6a² ÷ a³ = 6/a
For a Sphere (radius = r):
- Surface Area = 4πr²
- Volume = (4/3)πr³
- SA:V = 4πr² ÷ ((4/3)πr³) = 3/r
Rectangular Prism (length = l, width = w, height = h):
- Surface Area = 2(lw + lh + wh)
- Volume = l × w × h
- SA:V = 2(lw + lh + wh) ÷ (l × w × h)
Let’s calculate for a rectangular box with:
- Length (l) = 5 cm
- Width (w) = 3 cm
- Height (h) = 2 cm
Surface Area = 2(lw + lh + wh)
= 2(5×3 + 5×2 + 3×2)
= 2(15 + 10 + 6)
= 2(31)
= 62 cm²Volume = l × w × h
= 5 × 3 × 2
= 30 cm³SA:V Ratio = 62/30 = 2.07 cm^-1
Surface Area to Volume Ratios for Different Shapes
Sphere
For a sphere with radius r:
- Surface Area = 4πr²
- Volume = (4/3)πr³
- SA:V Ratio = 3/r
Cuboid
For a cuboid with length l, width w, and height h:
- Surface Area = 2(lw + lh + wh)
- Volume = l × w × h
- SA:V Ratio = 2(lw + lh + wh)/(l × w × h)
Rectangular Prism
- Surface Area = 2(lw + lh + wh)
- Volume = l × w × h
- SA:V Ratio = 2(lw + lh + wh)/(l × w × h)
Cylinder
For a cylinder with radius r and height h:
- Surface Area = 2πr² + 2πrh
- Volume = πr²h
- SA:V Ratio = (2πr² + 2πrh)/(πr²h)
References
- National Institute of Standards and Technology (NIST): https://www.nist.gov/
- Mathematics LibreTexts: https://math.libretexts.org/
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