Calculate the surface area of a sphere from radius, diameter, volume, or circumference. Also get hemisphere SA, spherical cap area, and volume with reference tables.
The surface area of a sphere is one of the most elegant results in geometry: SA = 4πr². This means the outer surface of a sphere is exactly four times the area of its great circle — a relationship Archimedes famously proved over two thousand years ago.
Knowing a sphere's surface area is essential in many fields. In physics, it determines the intensity of radiation or gravitational fields that fall off as 1/r² — a direct consequence of the 4πr² surface through which those fields spread. In engineering, the surface area governs heat transfer rates, material costs for coatings, and the amount of paint needed to cover tanks or domes. In biology, cell surface-area-to-volume ratios explain why cells must remain small to exchange nutrients efficiently.
This calculator goes beyond the basic formula. You can start from any measurement — radius, diameter, volume, or circumference — and it derives all the others. It also computes hemisphere surface areas (both the curved dome and the total including the base), and spherical cap areas for partial spheres cut at a given height. The cap formula, SA_cap = 2πrh, is invaluable for dome construction and lens design.
Presets cover everyday objects (ping-pong balls, baseballs, soccer balls) and astronomical bodies (Moon, Earth, Sun), making it easy to explore how surface area scales across vastly different sizes. A reference table and visual bar chart help you compare values at a glance.
This calculator is useful whenever the radius is not the quantity you start with. In real problems you may know a ball's diameter, a tank's circumference, or the volume of a spherical container and still need its coating area. Switching between those inputs by hand requires rearranging formulas and keeping track of unit powers, which is where small algebra mistakes often appear.
It is also helpful for partial-sphere problems. If you are comparing a full sphere, a hemisphere, and a spherical cap, this tool shows those values side by side so you can estimate paint coverage, dome cladding, or membrane area without reworking the geometry each time.
Surface Area: SA = 4πr² Volume: V = (4/3)πr³ Diameter: d = 2r Circumference: C = 2πr Great Circle Area: πr² Hemisphere Total SA: 3πr² (curved 2πr² + base πr²) Spherical Cap SA: 2πrh Cap Volume: (πh²/3)(3r − h) Radius from SA: r = √(SA / 4π) Radius from Volume: r = ∛(3V / 4π)
Result: Surface Area ≈ 1256.6371 cm², Volume ≈ 4188.7902 cm³, Cap Surface Area ≈ 251.3274 cm²
Choose Radius mode and enter 10 cm. The calculator uses SA = 4πr² = 4π(10²) = 400π ≈ 1256.6371 cm² and V = (4/3)πr³ = (4/3)π(1000) ≈ 4188.7902 cm³. With capHeight = 4 cm, the spherical cap area is 2πrh = 2π(10)(4) = 80π ≈ 251.3274 cm², and the great-circle area remains πr² ≈ 314.1593 cm².
The formula 4πr² packages several useful ideas into one expression. It tells you that the entire surface of a sphere depends only on the radius, and it scales with the square of that radius. If you double the radius, the surface area becomes four times larger. This is why even modest increases in diameter can dramatically change the amount of coating, fabric, or insulation needed for spherical objects.
Many real objects are only part of a sphere. Domes behave like hemispheres, and lenses or tank ends often behave like spherical caps. Those shapes do not use the same surface area as a full sphere, so it helps to break them out explicitly. This calculator shows the curved hemisphere area, the total hemisphere area including the base circle, and cap area from a chosen height so you can model partial-surface jobs accurately.
Radius is the cleanest starting point, but it is not always the measurement you are given. Manufacturers often publish diameter, field problems may provide circumference, and some science problems begin with volume. Converting all of those back to radius before finding surface area is standard practice, but it is easy to lose a factor of 2 or π during the rearrangement. Using the matching input mode reduces those conversion mistakes and lets you verify each relationship quickly.
SA = 4πr², where r is the radius. For a sphere of radius 5 cm, SA = 4π(25) ≈ 314.16 cm².
r = √(SA / 4π). For SA = 1000 cm², r = √(1000 / 12.566) ≈ 8.92 cm.
The curved part is 2πr². Including the flat circular base, the total is 3πr². For r = 10, total = 300π ≈ 942.48.
A spherical cap is the portion cut off by a plane. Its lateral surface area is 2πrh, where h is the cap height. This formula is independent of the base circle radius.
Smaller objects have higher SA:Vol ratios, meaning more surface per unit volume. This is why cells stay small (nutrient exchange) and why crushed ice melts faster.
The sphere has the smallest surface area for a given volume. A cube enclosing the same volume has about 24% more surface area.