The surface area of a sphere measures its curved outer surface, like the skin of an orange. It does not measure the space inside. Every point on the spherical surface lies the same distance r from the centre; this distance is the radius.
Which formula gives the whole surface area?
For radius r and diameter d, use:
The diameter is twice the radius. The second form follows by replacing r with half of d. Both formulas give square units: a radius in centimetres produces an area in square centimetres. The constant π is exact; 3.14 is only an approximation.
A sphere with radius 3 centimetres
Square the radius first, then multiply by 4π. For r = 3 cm, the exact area is 36π square centimetres. Rounded to two decimal places, the result is:
Working from a sphere's 6 cm diameter
A sphere has diameter 6 cm. Find its entire curved surface area. The formula needs the radius, so the diameter cannot be substituted directly for r.
Halve the diameter to obtain the 3 cm radius.
Square the radius before multiplying by the remaining factors. This produces a quantity in square centimetres.
Multiply by 4π. Keep the exact multiple of π, then round the final decimal to two places.
The surface area is exactly 36π square centimetres, approximately 113.10 square centimetres. This measures the outside surface, not the sphere's volume.
If the given measurement were a diameter of 6 cm, the answer would be the same. Treating that diameter as a radius would give four times the correct area. Keep π in the calculation until the final rounding step.
What is the area of a spherical lune?
A lune is the surface strip between two semicircles of great circles meeting at the same two opposite points. Its area is the same fraction of the total surface as its angle is of a full turn. Use α for degrees or θ for radians:
These two expressions use different angle units. For a 90° lune, take one quarter of the sphere's area. On the radius-3 sphere, that is 9π square centimetres. A surface lune is different from the three-dimensional wedge behind it.
Spherical zones and caps use vertical height
A zone is the part of the surface between two parallel cutting planes. A cap is the part beyond one cutting plane. Both curved areas can be calculated using the sphere's radius r and the relevant vertical height h:
For a zone, h is the perpendicular distance between the planes. For a cap, it runs from the cutting plane to the top of the sphere. It is not the sloping distance along the surface, and r is not the radius of the circular cut.
With sphere radius 3 cm and cap height 1 cm, the curved cap area is 6π square centimetres. This excludes the flat circular face that would close the cut solid. A hemisphere has height r, so its curved area is half the sphere's surface; add the base disk only if the problem asks for a closed hemisphere.
Surface area and volume answer different questions
Surface area concerns covering or coating the outside; volume concerns capacity or occupied space. Their units differ, and their formulas cannot be interchanged. All formulas here assume an ideal sphere, not an ellipsoid or an irregular rounded object. For the flat circular shape, see area of a circle.
