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(wherebis the base length andlis 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 Volume**

This 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 cm²Volume= (2 cm)³ = 8 cm³SA:V Ratio= 24/8 = 3 cm^-1

**Ellipsoid**

Surface Area: Approximation usingS ≈ 4π((a^p(whereb^p + a^pc^p + b^p * c^p)/3)^(1/p)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(wherebis the base length ands₁, s₂, s₃are 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= 6a²Volume= a³SA:V= 6a² ÷ a³ = 6/a

### For a Sphere (radius = r):

Surface Area= 4πr²Volume= (4/3)πr³SA:V= 4πr² ÷ ((4/3)πr³) = 3/r

### Rectangular Prism (length = l, width = w, height = h):

Surface Area= 2(lw + lh + wh)Volume= l × w × hSA: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 cm²

**Volume** = l × w × h

= 5 × 3 × 2

= 30 cm³

SA: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= 4πr²Volume= (4/3)πr³SA: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 × hSA:V Ratio= 2(lw + lh + wh)/(l × w × h)

### Rectangular Prism

Surface Area= 2(lw + lh + wh)Volume= l × w × hSA:V Ratio= 2(lw + lh + wh)/(l × w × h)

### Cylinder

For a cylinder with radius r and height h:

Surface Area= 2πr² + 2πrhVolume= πr²hSA: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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