📏 Enter known values

📐 Accepted units: mm, cm, dm, m, dam, hm, km, in, ft, yd, mi

Formula Reference

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Worked example

Data
Radius
\(b = {\color{#6c5ce7}{3\,\text{cm}}}\)
Area
\(a = \,?\)
Formula
\[a = 4\cdot\pi\cdot{{b}}^{2}\]
Steps
1Substitute the values
\[a = 4\cdot\pi\cdot{\left({\color{#6c5ce7}{3\,\text{cm}}}\right)}^{2}\]
Each letter is replaced by its value.
2Square
\[a = 4\cdot\pi\cdot9\,\text{cm}^{2}\]
\(3 \times 3 = 9\)\(\text{cm} \times \text{cm} = \text{cm}^{2}\)
3Multiply
\[a = 4\cdot\pi\cdot9\,\text{cm}^{2} \approx 113.0973\,\text{cm}^{2}\]
\(4\cdot\pi \approx 12.5664\)\(12.5664\cdot9 \approx 113.0973\)
Area: 113.0973 cm²
Check

If we use the result to calculate Radius, we should get the value you entered:

\[b = \sqrt{\frac{{a}}{4\cdot\pi}}\]
\[b = \sqrt{\frac{113.0973\,\text{cm}^{2}}{4\cdot\pi}} \approx 3\,\text{cm}\]
It gives 3 cm, the same as the value entered.
Calculate the Area
Please fill in the fields:
Radius
And leave empty
Area
Calculate the Radius
Please fill in the fields:
Area
And leave empty
Radius

How to calculate the surface area of a sphere

The surface area of a sphere is four times the area of a circle with the same radius:

\[ \text{Area} = 4\pi r^2 \]

Archimedes proved it by showing that the sphere has the same surface as the side of the cylinder that wraps it exactly. With the diameter \( d \), the formula becomes \( \text{Area} = \pi d^2 \).

To find the radius from the area:

\[ r = \sqrt{\frac{\text{Area}}{4\pi}} \]

How to use the calculator

  1. Enter the radius to get the area, or the area to get the radius.
  2. Leave the other field empty.
  3. Press Calculate. If you know the diameter, divide it by 2.

Worked examples

1. Sphere with a radius of 5 cm

\[ \text{Area} = 4\pi \times 5^2 = 100\pi \approx 314.16\ \text{cm}^2 \]

2. Radius of a sphere with an area of 200 m²

\[ r = \sqrt{\frac{200}{4\pi}} \approx 3.99\ \text{m} \]

3. Surface of a soccer ball

A size 5 ball has a circumference of about 69 cm, so its radius is \( 69 \div 2\pi \approx 10.98 \) cm and its surface is \( 4\pi \times 10.98^2 \approx 1{,}515\ \text{cm}^2 \), about 0.15 m².

4. Half a sphere (a dome)

A hemispherical dome with a radius of 6 m has a curved surface of \( 2\pi \times 6^2 \approx 226.19\ \text{m}^2 \). If it also has a floor, add the circle: \( \pi \times 6^2 \approx 113.1\ \text{m}^2 \).

How the area grows with the radius

If the radius doubles, the surface is multiplied by four; if it triples, by nine. The Earth, with a mean radius of about 6,371 km, has a surface of about \( 4\pi \times 6{,}371^2 \approx 510 \) million km².

RadiusDiameterArea
1212.57
2450.27
36113.1
48201.06
510314.16
612452.39
714615.75
816804.25
9181,017.88
10201,256.64

When Do You Need to Calculate the Surface Area of a Sphere?

⚽ Balls and Sports Equipment

The surface of a ball sets how much leather, rubber or synthetic panel is needed to make it. Manufacturers also use it for printing and coatings.

Material estimates for making balls
🛢️ Spherical Tanks

Gas and liquid storage tanks are often spherical because the shape spreads the pressure evenly. The surface area gives the steel plate and the paint needed.

Plate and paint quantities for tanks
🏛️ Domes and Planetariums

Domes are half spheres. Their curved surface tells how much roofing, insulation or projection screen they need.

Cover and finish domes
🌍 Astronomy and Geography

The surface of a planet or a star is calculated as a sphere. It is used for the area of the Earth, the radiation emitted by stars and climate models.

Work with planetary measurements
💊 Medicine and Pharmacy

The surface of pills, beads and cells affects how fast a drug dissolves or a substance is absorbed. Smaller spheres have more surface per unit of volume.

Understand dissolution and absorption
🎈 Balloons and Decorations

The surface of a balloon or a spherical lamp sets how much material or paint covers it.

Plan coverings and decorations

Common Mistakes

⚠️ Forgetting the Factor 4
Common Error: Using πr², which is the area of a flat circle, not of the whole sphere.
✅ Solution: Area = 4πr²
⚠️ Using the Diameter as the Radius
Common Error: Entering the diameter as r makes the area four times too large.
✅ Solution: r = d ÷ 2, or use Area = πd²
⚠️ Confusing Area and Volume
Common Error: Using 4/3 πr³, which is the volume, when the question asks for the surface.
✅ Solution: Surface → 4πr² (square units); volume → 4/3 πr³ (cubic units)
⚠️ Squaring After Multiplying
Common Error: Calculating (4πr)² instead of 4π × r². For r = 5 this gives about 3,948 instead of 314.16.
✅ Solution: Square only the radius
⚠️ Hemisphere Without Its Base
Common Error: For a solid hemisphere, using only 2πr² and forgetting the flat circle.
✅ Solution: Closed hemisphere: 2πr² + πr² = 3πr²
⚠️ Using the Circumference as the Diameter
Common Error: Wrapping a tape around a ball and using that number as the diameter.
✅ Solution: Diameter = circumference ÷ π

Applications by Industry

Manufacturing
  • Balls: Material for sports balls
  • Bearings: Surface finish of steel balls
  • Tanks: Steel plate for spherical pressure vessels
Earth & Space Science
  • Planets: Surface area of planets and moons
  • Stars: Luminosity from the radius and the temperature
  • Climate: Energy received per unit of the Earth's surface
Pharmacy & Biology
  • Drug release: Dissolution of spherical tablets and beads
  • Cells: Surface-to-volume ratio of cells
  • Nanoparticles: Reactive surface of spherical particles
Architecture
  • Domes: Roofing and insulation for hemispherical domes
  • Geodesic domes: Panels needed to cover a dome
  • Water towers: Paint for spherical tanks
Chemistry & Physics
  • Droplets: Surface tension and evaporation of drops
  • Catalysis: Active surface of spherical catalysts
  • Radiation: Intensity spread over a sphere: the inverse-square law
Education
  • Geometry: Surface of solids of revolution
  • History: Archimedes and the sphere inside a cylinder
  • Scaling: How the surface grows with the square of the radius

External Resources

Other calculators

Area of a Quadrangular Prism
Area of a Quadrangular Prism
Area of a Rectangle
Area of a Rectangle
Area of a Hexagon
Area of a Hexagon
Area of a Triangle
Area of a Triangle
Area of a Rhombus
Area of a Rhombus
Circle Calculator
Circle Calculator
Surface Area of a Cylinder
Surface Area of a Cylinder
Area of a Cube
Area of a Cube

Geometry Calculators

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