Sphere Surface Area Calculator

Radius (r)

Diameter (d)

Surface area (A)

Volume (V)

Make it a hollow shell (add an inner radius)

Enter the inner radius to get the both-face surface area, the wall thickness, and the material volume of a spherical shell.

Estimate a coating or material cost

Enter a price per square metre to estimate the cost of painting or coating the surface.

Enter any one measurement, the radius, diameter, surface area, or volume, and the calculator fills in the other three for you.

Please input values into the fields

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Show how the surface area scales with the radius

A bar chart of how the surface area changes as the radius grows.

This sphere surface area calculator allows you to calculate not only the outer surface of a sphere but also much more. Just enter one of the available measurements (radius, diameter, surface area, volume), and the tool will calculate the rest for you. By entering a value and selecting the unit, other values are automatically calculated with the surface area being shown as the main result. There are also options for hollow shells and calculating coating costs.

What is the surface area of a sphere?

A sphere is a three-dimensional object that has the perfect round shape. Every point on its surface is at an equal distance from its center and it does not have any edges or corners; instead, it only has a smooth curved surface. Balls, soap bubbles, planets, and water droplets are typically shaped like spheres. The surface area is the total area of this curved surface and is expressed in square units such as square centimeters or square feet. Since a sphere is uniquely defined by just one value - its radius - all other measurements can be derived from it.

How to use this calculator:

Enter one of the available measurements (radius, diameter, surface area, volume) into its corresponding field and select the unit. After entering a value, the tool will automatically calculate the remaining values and display the surface area as the main result. It is not necessary to specify the type of measurement entered. To start with another measurement, simply clear the previously filled-in field and enter a new value. By activating the hollow shells option, you can model a hollow sphere, and by activating the cost option, you can estimate the coating costs.

The formula for calculating the surface area of a sphere:

The surface area is determined only by the radius. In the formula, you multiply the number π with the square of the radius and quadruple the result.

A = 4 \pi r^2

In this formula A is the surface area, r is the radius and pi is approximately 3.14159. If you know the diameter, an easier way to express the same relationship is because the diameter is twice the radius.

A = \pi d^2

Let's say that the radius of a sphere is 9 centimeters. You square the radius which gives you 81, multiply it by four and then by pi.

A = 4 \pi \times 9^2 = 324\pi \approx 1017.88 \ \text{cm}^2

Conversely, one can calculate the radius from the surface area by dividing the surface area by four times pi and then taking the square root.

r = \sqrt{\frac{A}{4\pi}}

The surface can be directly calculated from the volume.

It is not necessary to determine the radius first. Since surface area and volume are connected by a simple identity, one can calculate directly from the other. If you take the surface area cubed, you get the product of the square of the volume and 36 multiplied by pi. After simplification, this results in a formula that uses a cubic root.

A^3 = 36\pi V^2 \quad\Longrightarrow\quad A = \sqrt[3]{36\pi V^2}

This calculator uses this relationship directly so that you can calculate the surface area by simply inputting the volume without having to do any intermediate calculations.

All measurement values of a ball at one place.

Every measurement of a sphere is related to the radius in some way. This table summarizes the relationships used by this calculator so that when you enter one value, the others can be calculated for you.

Measurement	Symbol	Formula from the radius
Diameter	d	d = 2r
Great-circle circumference	C	C = 2 pi r
Cross-section (great-circle) area	a	a = pi r squared
Surface area	A	A = 4 pi r squared
Volume	V	V = four-thirds pi r cubed
Radius from surface area	r	r = square root of (A / 4 pi)

Examples of calculations:

The following table shows the surface area for some common radii. The results are rounded to two decimal places. Please note that as the radius increases, the surface area grows quickly. This is because the radius must be squared.

Radius	Calculation	Surface area
4 cm	4 pi times 4 squared	201.06 sq cm
9 cm	4 pi times 9 squared	1017.88 sq cm
12 cm	4 pi times 12 squared	1809.56 sq cm
20 cm	4 pi times 20 squared	5026.55 sq cm

When calculating with the diameter, you must first halve the diameter to get the radius. The method is then identical. If the diameter is 16 centimeters, the radius is 8, so the surface area is equal to four times pi multiplied by 8 squared, which is approximately 804.25 square centimeters. With this calculator, you can simply enter the value 16 in the field for the diameter to read off the surface area directly.

Why this relationship to the square of the radius is important:

Since the radius is squared, doubling the radius does not double the surface area; it quadruples it. Tripling the radius will increase the surface area nine-fold. This nonlinear scaling explains why a beach ball has so much more surface area than a marble even though the size difference between them might not seem that great. An enlarged plot in the analysis section shows this directly.

\frac{A(kr)}{A(r)} = \frac{4\pi (kr)^2}{4\pi r^2} = k^2

The difference between surface and volume:

Surface area and volume are easy to confuse with each other. The surface area is a two-dimensional outer shell, while the volume is a three-dimensional interior space. For a sphere, both quantities are related by a simple ratio.

\frac{A}{V} = \frac{3}{r}

This ratio shows that the surface area is larger in proportion to the volume for smaller spheres. This fact can explain many phenomena in the real world. Small water droplets and cells are able to exchange heat or gases quickly because their surface area makes up a large portion of them. Smaller pieces of ice melt faster than one big chunk of ice with the same weight. Very small droplets of fuels burn more easily than larger amounts of the same fuel.

Total surface area, curved surface area, hemisphere.

For a complete sphere there is no difference between the total surface area and the curved surface area. The entire curved surface area, calculated as four times pi multiplied by the square of the radius, is the total surface area. When you cut a sphere in half, however, there is a difference. The curved surface area for one hemisphere is two times pi multiplied by the square of the radius, and then there's also a circular flat bottom with an area of pi multiplied by the square of the radius, making the total surface area three times pi multiplied by the square of the radius.

A_{\text{hemisphere}} = 2\pi r^2 + \pi r^2 = 3\pi r^2

Hollow balls and shells

Real spheres, tanks and bubbles usually have a wall thickness and surfaces on both the inside and outside. By enabling the "shell" option and entering an inner radius you can calculate the total area of both sides, the wall thickness and volume of material within the walls. The total surface area of a shell is the sum of the inner and outer areas.

A_{\text{shell}} = 4\pi \left(r_{\text{out}}^2 + r_{\text{in}}^2\right)

Applications for calculating the surface area of a sphere

Because spherical structures are found everywhere, this calculation is used in many fields. Painters and coating specialists use the surface area calculation to determine how much paint they need for domes, tanks, and buoys. Engineers estimate heat loss from boilers and furnaces which is proportional to surface area. Atmospheric scientists study the Earth's surface. The surface area of the Earth is about 510 million square kilometers with an average radius used as a basis for calculations of about 6371 km. Biologists compare the ratio of surface area to volume in cells and droplets to determine how fast nutrients are absorbed or gases exchanged.

Tips for accurate results:

Measure the radius exactly from the center or measure the diameter of the widest part and input half that value into the calculator. If you mistake the diameter for the radius, it will give a surface area four times larger than the correct value. Make sure your units match. If the radius is in centimeters, then the surface area will be in square centimeters unless you change the output unit manually. For irregularly shaped objects use an average radius. Note that the surface area changes proportionally to the square of the radius so small errors in the radius can lead to larger errors in the area.

Frequently asked questions

What is the formula for calculating the surface area of a sphere?

The surface area is given by A = 4 times pi, multiplied by the square of the radius. As Pi is approximately 3.14159, the surface area with a radius of 9 is about 1017.88 units squared.

How to calculate surface area from diameter or volume?

To calculate using diameter, use the formula A = πr2.*(Diameter)² . Halving the diameter will give you the same result as 4πr². When calculating from volume, use the formula A = cube root of (36V/π).*π * (Volume)²). Regardless of what value you input, this calculator will automatically and instantly perform both calculations.

If the radius is doubled, how does that change the surface area?

The surface area is not halved but quadrupled. This is because it depends on the square of the radius. If the radius is tripled, then the surface area becomes nine times larger.

What is the difference between total surface area and curved surface area?

For a complete sphere there is no difference between the two surfaces. This is because the entire surface area is described by one curved surface with the formula 4 pi r squared. A distinction is only required for hemispheres. The total surface area of a hemisphere consists of a flat circular base with an area of 3 pi r squared and a curved surface with an area of 2 pi r squared.

Can you calculate the radius or volume if only the surface area is known?

Yes. The radius is equal to the value obtained by dividing the square root of the surface area by 4π, and the volume can be calculated from it. If you enter the surface area in the corresponding field, the calculator will automatically fill in the radius, diameter and volume.

Related calculators

Surface Area CalculatorFind the surface area of a sphere, cube, box, cylinder, cone, pyramid, capsule, spherical cap, frustum, or ellipsoid. Pick a solid, enter its measurements, and get the surface area and volume in any unit.

Cylinder Surface Area CalculatorEnter any two of a cylinder's radius, height, volume, or lateral surface area and get the total and lateral surface area, volume, and more instantly, with full unit conversion.

Cube Surface Area CalculatorEnter any one of a cube's edge, total or lateral surface area, volume, or space diagonal and get all the others instantly, with full unit conversion.

Hemisphere Surface Area CalculatorEnter any one of a hemisphere's radius, diameter, total or curved surface area, or volume and get all the others instantly, with full unit conversion.

Cone Surface Area CalculatorFind the lateral, base, and total surface area of a cone from any two of its radius, height, or slant height, plus its volume and apex angles.

Volume CalculatorFind the volume of a sphere, cube, box, cylinder, cone, pyramid, capsule, spherical cap, frustum, ellipsoid, or tube. Pick a solid, enter its measurements, and get the volume and surface area in any unit.

Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Wikipedia: Sphere
Definition, surface area and volume formulas, and properties of the sphere.
Wikipedia: Surface area
How surface area is defined and computed for common solids.
Math is Fun: Sphere
Plain-language explanation of a sphere's radius, surface area, and volume.