How do you find the volume and surface area of a sphere?

The volume of a sphere is V = (4/3)πr³ and the surface area is SA = 4πr², where r is the radius. Both formulas require only the radius. For a sphere with radius 5: • Volume: V = (4/3) × π × 5³ = (4/3) × π × 125 = 523.5988 cubic units • Surface area: SA = 4 × π × 5² = 4 × π × 25 = 314.1593 square units • Diameter: d = 2 × 5 = 10 units Memory aids: • Volume: think '4/3 pi r cubed' — the cube tells you it is cubic units • Surface area: think '4 pi r squared' — four times the area of the great circle • SA = 4 × (area of great circle) is a beautiful geometric fact — the surface area exactly equals four of the circular cross-sections through the centre • This calculator solves from any of the four properties: radius, diameter, volume, or surface area

How do you find the radius from the volume or surface area?

To find radius from volume: rearrange V = (4/3)πr³ → r = ∛(3V/4π). To find radius from surface area: rearrange SA = 4πr² → r = √(SA/4π). To find radius from diameter: r = d/2. Worked examples: • From V = 100: r = ∛(3×100/(4π)) = ∛(23.8732) = 2.8794 units • From SA = 100: r = √(100/(4π)) = √(7.9577) = 2.8209 units • From d = 14: r = 7 units Order of operations matters: • Volume → radius: compute 3V/(4π) first, then take cube root (∛) • Surface area → radius: compute SA/(4π) first, then take square root (√) • This calculator handles all four solve directions — select what you know and enter it

What is a great circle of a sphere?

A great circle is the largest possible circle that can be drawn on a sphere — it has the same radius as the sphere. Any plane passing through the centre of a sphere intersects it in a great circle. The equator of Earth is a great circle. Great circles are the shortest paths between two points on a sphere (geodesics), which is why aircraft fly great circle routes. Great circle properties: • Radius = sphere radius r • Area = πr² (same as a flat circle of radius r) • Circumference = 2πr • Surface area of sphere = 4 × great circle area (exactly four great circles) • A sphere can be divided into any number of great circles, all passing through a common pair of antipodal points Practical uses: • Aviation: great circle routes are the shortest flight paths • Navigation: early navigators used great circle sailing • Geometry: great circles play the role of 'straight lines' in spherical geometry

How does a sphere's surface area compare to its volume?

The surface area-to-volume ratio (SA:V = 4πr²/(4πr³/3) = 3/r) decreases as the sphere grows. A larger sphere has proportionally less surface area relative to its volume. This ratio has profound biological and physical implications. SA:V ratio = 3/r: • r=1: SA=12.566, V=4.189, ratio=3.0 • r=2: SA=50.265, V=33.510, ratio=1.5 • r=5: SA=314.159, V=523.599, ratio=0.6 • r=10: SA=1256.637, V=4188.790, ratio=0.3 Why this matters: • Cells: small cells have high SA:V — efficient for nutrient/waste exchange; large cells struggle • Cooling: small spheres cool faster (more surface per unit volume) • Strength: bubbles minimise SA:V — sphere is the shape with minimum surface area for a given volume • Planets: larger planets retain internal heat longer (lower SA:V) • Engineering: heat exchangers maximise SA:V for efficient transfer

What is the relationship between a sphere and its circumscribed cube?

A circumscribed cube (the smallest cube containing the sphere) has side length equal to the diameter of the sphere: a = d = 2r. A inscribed cube (largest cube fitting inside the sphere) has side length a = d/√3 = 2r/√3. Key relationships: • Circumscribed cube side: a = 2r (cube just touches sphere at 6 face centres) • Inscribed cube side: a = 2r/√3 ≈ 1.1547r (all 8 vertices on sphere surface) • Volume of sphere / circumscribed cube = (4/3)πr³ / (2r)³ = π/6 ≈ 52.36% • Surface area of sphere / circumscribed cube surface = 4πr² / 24r² = π/6 ≈ 52.36% • Same ratio! A sphere fills exactly π/6 of its circumscribed cube by both volume and SA This was proven by Archimedes (~250 BC), who also showed that a sphere inscribed in a cylinder fills exactly 2/3 of the cylinder — a result he considered his greatest discovery.

Sphere Calculator — Volume, Surface Area, Diameter & More

The Sphere Calculator computes all properties of a sphere from any single known measurement. Four input modes: radius, diameter, volume, or surface area. Enter any one value and the calculator instantly finds all four properties plus the great circle area and great circle circumference. The surface ...

SPHERE CALCULATOR

SOLVE FROM

Radius (r)

Must be greater than 0

RADIUS (r)

units

DIAMETER (d)

units

VOLUME (V)

523.598776

units³

SURFACE AREA (SA)

314.159265

units²

Great Circle Area

78.539816

units²

Great Circle Circum

31.415927

units

SA / V ratio0.6

V / SA ratio1.6667

SA = 4 × great circle78.54 × 4 = 314.1593

Circumscribed cube side5.7735

STEP-BY-STEP

Given: radius r = 5

Diameter: d = 2r = 10

Volume: V = (4/3)πr³ = (4/3) × π × 5³ = 523.598776

Surface area: SA = 4πr² = 4 × π × 5² = 314.159265

Great circle area: A = πr² = 78.539816

QUICK EXAMPLES

LIVE DIAGRAM

EXACT EXPRESSIONS

Volume

(4/3)π × 5³ = (4/3)π × 125

Surface area

4π × 5² = 4π × 25

Great circle

π × 5² = 78.539816

SCALE COMPARISON

Radius 5

Diameter 10

Great C 31.416

Created with❤️byeaglecalculator.com

SPHERE FORMULAS

ALL SPHERE FORMULAS

NAME	FORMULA	DESCRIPTION
Volume	V = (4/3)πr³	Four thirds pi r cubed
Surface area	SA = 4πr²	Four times the great circle area
Diameter	d = 2r	Twice the radius
Radius from volume	r = ∛(3V/4π)	Cube root of 3V over 4π
Radius from surface area	r = √(SA/4π)	Square root of SA over 4π
Great circle area	A = πr²	Area of cross-section through centre
Great circle circumference	C = 2πr	Circumference of equator
SA:V ratio	SA/V = 3/r	Decreases as sphere grows
Circumscribed cube	a = 2r (side length)	Sphere fills π/6 ≈ 52.36% of cube

HOW TO USE

1
Select what you know: radius (r), diameter (d), volume (V), or surface area (SA). The calculator solves all other properties from any single input.
2
Enter your value. The calculator accepts any positive number — try the Earth example (r=6371 km) to see how the formulas scale.
3
All four primary properties appear instantly, plus great circle area and circumference, SA:V ratio, and circumscribed cube side length.
4
Read the step-by-step solution to see exactly which formula was used and how the radius was derived from your input.
5
Use the quick example buttons to load common sphere sizes including a unit sphere (r=1), Earth (r=6371), and known-volume and known-surface-area cases.

WORKED EXAMPLE

Example 1 (radius r=5): V=(4/3)π×125=523.599, SA=4π×25=314.159, d=10, great circle area=78.540. Example 2 (from volume V=100): r=∛(300/4π)=∛(23.873)=2.879. Example 3 (from SA=314.159): r=√(314.159/4π)=√25=5. Earth r=6371km: V=1.083×10¹² km³, SA=5.101×10⁸ km².

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Last updated: April 29, 2026 · Formula verified by EagleCalculator team · Eagle-eyed accuracy for every calculation.