What this calculator does
A hemisphere is exactly half of a sphere, cut through its centre by a flat plane. This calculator takes a single measurement — the radius — and returns the three surface-area figures people actually need: the curved (lateral) surface area of the dome, the base area of the flat circular cut, and the total surface area. Choose whether the shape is solid (closed) or hollow (open, like a bowl), pick your length unit, and every figure updates instantly in that unit squared.
It is a close cousin of a circle-area problem: the base of a hemisphere is a full circle, and the curved part is half a sphere. If you are comfortable turning one measurement into a proportion or a share, our percentage calculator and fraction calculator handle the everyday arithmetic that often sits alongside a geometry answer.
The formulas
Every figure comes from the radius r and the constant π (≈ 3.14159):
Curved (lateral) surface area = 2 · π · r²
Base (flat circular) area = π · r²
Total surface area (solid) = 2·π·r² + π·r² = 3 · π · r²
Total surface area (hollow) = 2 · π · r² (curved only)
The key relationship
Worked example: a 5-metre solid hemisphere
Take a solid hemisphere with a radius of 5 m. Squaring the radius and applying each formula gives the numbers below — computed by the same engine that powers the calculator, so they always match what you see above.
| Step | Formula | Value |
|---|---|---|
| Radius squared | r² = 5 × 5 | 25 m² |
| Curved surface | 2 × π × r² | 157.08 m² |
| Base disk | π × r² | 78.54 m² |
| Total (solid) | 3 × π × r² = curved + base | 235.62 m² |
Because the base disk (78.54 m²) is exactly half the curved surface (157.08 m²), it makes up one third of the 235.62 m² total — a useful sanity check for any radius.
Solid vs hollow: which total do you want?
The only difference between the two options is whether the flat base is counted:
- Solid (closed) hemisphere — a cast dome, a paperweight, a half-ball. The total is the curved surface plus the base disk: 3·π·r².
- Hollow (open) hemisphere — a mixing bowl, a dome roof, a colander shell. There is no bottom, so the total is the curved surface only: 2·π·r². The calculator still reports the base area separately in case you need it.
Curved surface area and base area are identical in both cases — only the total changes.
How area grows with the radius
Surface area scales with the square of the radius, so doubling the radius quadruples every figure. The reference table makes the pattern obvious:
| Radius (r) | Curved 2πr² | Base πr² | Total (solid) 3πr² |
|---|---|---|---|
| 1 | 6.28 | 3.14 | 9.42 |
| 2 | 25.13 | 12.57 | 37.7 |
| 5 | 157.08 | 78.54 | 235.62 |
| 10 | 628.32 | 314.16 | 942.48 |
| 20 | 2,513.27 | 1,256.64 | 3,769.91 |
Assumptions and limitations
- The shape is treated as a perfect (geometric) hemisphere cut exactly through the sphere's centre, so the base is a true great circle of radius r.
- A real hemispherical shell with wall thickness has separate inner and outer surfaces; this tool models an idealised zero-thickness surface.
- All lengths must be in the same unit; the result is reported in that unit squared. The unit selector only relabels the answer — it does not convert between units.
- The radius must be zero or positive. A radius of 0 gives an area of 0, and any negative entry is clamped to 0. The radius is capped at 1,000,000 units to keep the field sane.
Frequently asked questions
What exactly is a hemisphere?+
Take a sphere and slice it with one flat plane that passes through its centre point — each of the two resulting pieces is a hemisphere. It keeps the sphere's curved shell over exactly half its surface and gains one new flat, circular face where the slice was made.
What is the formula for the surface area of a hemisphere?+
A solid hemisphere's total surface area is 3*pi*r^2, made up of the curved (lateral) surface 2*pi*r^2 plus the flat circular base pi*r^2, where r is the radius.
What is the curved surface area of a hemisphere?+
The curved surface area is 2*pi*r^2 — exactly half of a full sphere's surface area (4*pi*r^2), since a hemisphere is half a sphere.
What is the difference between curved surface area and total surface area of a hemisphere?+
Curved surface area (2*pi*r^2) is only the domed outer surface. Total surface area (3*pi*r^2) adds the flat circular base — the great circle formed where the sphere was cut.
Does a hollow hemisphere (like a bowl) have a base area?+
No. A hollow or open hemisphere — such as a bowl or dome with no bottom — has only the curved surface, 2*pi*r^2. The base disk is only counted for a solid, closed hemisphere.
How do you find the radius from the surface area of a hemisphere?+
Rearrange the formula for the area you have. For total surface area: r = sqrt(A_total / (3*pi)). For curved surface area: r = sqrt(A_curved / (2*pi)).
How is the area of a hemisphere related to the volume?+
They use the same radius but are different measures: surface area (in square units) describes the outer skin, while volume, (2/3)*pi*r^3, (in cubic units) describes the space it encloses. This calculator reports area only.
What units does the hemisphere area calculator use?+
Enter the radius in millimeters, centimeters, meters, inches or feet. The area results are automatically reported in that unit squared (e.g. a radius in meters gives an area in square meters).
Why is the curved surface area of a hemisphere half a sphere's area, not a quarter?+
A sphere's full surface (4*pi*r^2) is symmetric top-to-bottom around any great circle, so cutting it exactly in half through the centre gives each hemisphere exactly half the curved surface, 2*pi*r^2 — not a quarter, which would only apply if you also sliced it another way.
What is a real-world example of hemisphere surface area?+
A dome roof, a mixing bowl, or the rounded top of a water tank are commonly modeled as hemispheres. For a dome you usually want curved surface area (the material needed to cover it); for a solid casting you'd want total surface area.
Does the surface area formula change for a very small or very large radius?+
No — 2*pi*r^2, pi*r^2 and 3*pi*r^2 hold for any radius greater than zero. Area simply scales with the square of the radius, so doubling the radius quadruples every area figure.
Can the radius be zero or negative?+
A radius of zero degenerates every area to zero (a point has no surface). A negative radius has no physical meaning, so the calculator clamps any negative entry to zero rather than returning a nonsensical result.
How accurate is this hemisphere area calculator?+
It uses the full double-precision value of pi and the exact closed-form formulas (2*pi*r^2, pi*r^2, 3*pi*r^2), so results are accurate to the precision shown for any radius within the input range.
Is "curved surface area" the same thing as "cap area"?+
Yes — curved surface area, lateral surface area and cap area all refer to the same domed portion of a hemisphere, 2*pi*r^2. This calculator uses "curved (lateral) surface area" throughout; treat any of these three labels as interchangeable when comparing figures across sources.
What is the circumference of a hemisphere's flat edge?+
The rim where the flat base meets the curved dome is a circle of radius r, so its circumference is 2*pi*r — the same great-circle circumference formula used for the sphere it was cut from. That's a length (in the chosen unit, not squared), useful for sizing a rim, gasket or trim rather than a covering material.
Disclaimer
Sources
- Wolfram MathWorld — Sphere
- Wikipedia — Sphere
- Math Open Reference — Surface Area of a Sphere
- Encyclopedia of Mathematics — Sphere
Formula and data last reviewed by the TheCalculatorHive team on 11 July 2026. Figures are for general information, not professional advice.
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