How to Use the Hemisphere Calculator
This calculator allows you to compute radius, height or volume of a hemisphere given a known value.
To compute the surface area and volume of a hemisphere in terms of its radius (or diameter, or circumference), click the 'Radius' button and enter the known value. The surface area and volume will appear in the both the sphere preview and the math area, where the formulae used to make the calculations are shown. To save either the calculated surface area or volume, click the button that shows the resulting value and it will be copied to the clipboard, or click the 'Save' button to store the calculation in the calculator tape for later reference.
To calculate a hemisphere's radius and volume in terms of its surface area, you can click the 'Area' button and supply the known value for the area. This will also display buttons with the calculated hemisphere volume and radius values that can be used to copy these calculations to the clipboard.
Finally, to calculate the radius and area in terms of the sphere's volume, click the 'Volume' button and enter the known volume.
What are Hemispheres?
A hemisphere is created when a plane passes through the origin of a sphere, dividing the sphere into two halves, each referred to as a hemisphere. The surface area of this shape includes the dome-shaped half sphere as well as the area of the circular base. Because a hemisphere is exactly half a sphere, the volume of a hemisphere is exactly half the volume of a sphere of corresponding radius.
Hemispheres are often created when the dividing plane passes through the sphere's origin orthogonal to the sphere's axis of rotation. In this case the plane is dividing the sphere at the equator, in effect creating a Northern Hemisphere and a Southern hemisphere from the original volume.
Hemispheres are also commonly referred to as subdivisions of the celestial sphere in astronomoy. The celestial sphere is a construct used to visualize the positions of astronomical objects as though they were mapped to the surface of a sphere being viewed by an observed placed at the sphere's origin.
Mathematical Properties of Hemispheres
Because a hemisphere is exactly a sphere divided into two halves, the formulae for dealing with hemispheres relate very succintly to the same calculations for spheres. For example, the volume of a hemisphere is calculated using a modified version of a sphere's volume formula, just divided by two.
Equation for Volume of a Hemisphere
There surface area of a hemisphere has a similar half proportion, although like cones, the total area of a hemisphere includes the area of the circular base. Because of this, we this hemisphere calculator finds the total area in two steps. First, the calculator finds the hemisphere's dome area, and second the calculator finds the base area. The calculator then adds these two values to get the total surface area.
Equation for Area of a Hemisphere
The radius or height can easily be derived from the volume of a hemisphere using the following equation...
Equation for Finding the Radius of a Hemisphere from Volume
You can see when a radius or height is not provided, this hemisphere calculator finds the radius (or height) of a hemisphere, and substitutes that value into the conventional equations for volume and surface area.