Surface Area of a Hemisphere Calculator

A hemisphere is half of a sphere cut by a plane through the center. The curved cap is the exterior of the spherical part. The base is the flat circle where the cut was made.

Curved surface area (CSA) counts only the dome-shaped shell: CSA = 2πr². Total surface area (TSA) adds the base circle: TSA = 3πr². Word problems may ask for one or the other, so read the prompt carefully.

You will see hemispheres in dome roofs, storage tank heads, bowls, and planet models. The same radius links every formula on this page and in the definition guide.

The curved part is exactly half of a full sphere surface (4πr²), which is why CSA uses 2πr². The base is a circle with area πr². Total surface area combines both pieces.

Radius r must use the same length unit throughout. If you know diameter, convert with r = d/2 before substituting. See the formula article for geometric interpretation.

Start with a clear sketch. Label radius, mark the flat base, and note whether the problem wants curved area only or total area including the base.

More scenarios appear in our examples article for construction, manufacturing, and education.

Use CSA when a problem mentions lateral area, curved area, or the surface of the spherical cap. Coating a dome exterior without the floor slab is a practical case.

Common mistakes include adding the base when the prompt excluded it, or using πr² (circle area) instead of 2πr². The curved surface guide walks through checks.

Choose TSA when you need the full exterior of a closed hemisphere shape, such as a solid model or a capped tank head with an exposed base ring.

Compare with CSA: TSA is always larger by exactly πr², the base area. Read total surface area explained for derivation notes.

Full sphere surface area is 4πr². A hemisphere curved cap is half of that: 2πr². A full sphere has no flat base; a hemisphere adds πr² when you need total surface area.

Volume also differs: sphere (4/3)πr³ versus hemisphere (2/3)πr³. Keep surface and volume questions separate to avoid mixing formulas on tests.

Radius r = d/2. Then CSA = 2π(d/2)² = (πd²)/2 and TSA = 3π(d/2)² = (3πd²)/4.

Enter diameter directly in the calculator and radius updates automatically. The diameter guide includes accuracy tips for measured diameters.

Use radius or diameter input, then read curved surface area, total surface area, base area, volume, and surface-to-volume ratio in one view.

Any field can drive the calculation. That helps when a drawing lists volume, a spec states total surface area, or you only measured diameter on site.

Examples sit directly under the input panel. For a full walkthrough of features, read the calculator guide.

Mixing curved and total surface area is the most frequent error. Underline whether the base counts before you substitute.

Using πr² for the curved cap is another slip. The cap is 2πr²; πr² is only the base.

Forgetting square units on the final answer loses marks in class and causes unit errors in engineering notes.

Applying sphere volume (4/3)πr³ to a hemisphere problem without adjusting to (2/3)πr³ confuses separate questions.

Volume V = (2/3)πr³ measures capacity inside the shape. Surface area measures the outer shell. Both appear in the calculator so you can compare packaging, fluid storage, and coating needs on one diagram.

Surface-to-volume ratio links the two: for a hemisphere, TSA/V = 9/(2r). Smaller radii have higher ratios, which matters in heat transfer and biology scale models at an introductory level.

Deeper coverage lives in volume and surface area of a hemisphere. Sphere geometry and unit conversion articles on the blog add context when you move beyond the first calculation.