Volume of a Hemisphere Calculator

A hemisphere is exactly half of a sphere, split perfectly along its central plane. The mathematical foundation relies primarily on the radius (r) of the circular base. The core formulas used by this calculator include: Volume ($V = \frac{2}{3}\pi r^3$), which is half the volume of a full sphere; Base Area ($A_{base} = \pi r^2$), representing the flat circular bottom; Calotte Area ($A_{calotte} = 2\pi r^2$), which is the curved dome surface; and Total Surface Area ($A_{total} = 3\pi r^2$), combining both the base and curved sections. When given alternative inputs like diameter or volume, the calculator rearranges these algebraic relationships to solve for the missing variables.

Let us calculate the volume and total surface area of a hemisphere with a given diameter of 12 centimeters. Step 1: Find the radius by dividing the diameter by two. Radius ($r$) = 12 / 2 = 6 cm. Step 2: Compute the volume using the formula $V = \frac{2}{3}\pi r^3$. Substituting our radius: $V = \frac{2}{3} \times \pi \times 6^3 = \frac{2}{3} \times \pi \times 216 = 144\pi \approx 452.39$ cubic centimeters. Step 3: Compute the total surface area using $A_{total} = 3\pi r^2$. Substituting the radius: $A_{total} = 3 \times \pi \times 6^2 = 3 \times \pi \times 36 = 108\pi \approx 339.29$ square centimeters. Thus, the hemisphere holds approximately 452.39 cm³ of volume.

Always verify whether your input measurement is a radius or a diameter before starting your calculations, as confusing the two will quadruple your final volume error. When dealing with physical containers like bowls or domes, remember that internal volume calculations must use internal dimensions, accounting for the thickness of the material walls. Finally, keep your unit consistency strict; if your radius is in meters, your calculated volume will automatically be in cubic meters.