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

$$V_{\mathrm{hemisphere}}=\frac{2}{3}\pi r^3$$

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

$$A_{\mathrm{dome}}=2\pi r^2$$ $$A_{\mathrm{base}}=\pi r^2$$ $$A_{\mathrm{total}}=A_{\mathrm{dome}}+A_{\mathrm{base}}$$

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

$$r={\left(\frac{3V}{2\pi }\right)}^{1/3}$$

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.

Hemisphere Calculator Definitions

Hemisphere

A sphere is a three dimensional shape made when a sphere is divided into two equal halves by a plane passing through the origin of the sphere.

Base

The circular bottom surface of a hemisphere.

Diameter

A sphere's diameter is a line segment that extends from one point on the outer edge of a hemisphere's base, through the center point and terminating on an antipodal point on the opposite side of the base. The measurement of a hemisphere's diameter is exactly twice its radius, and is show in the hemisphere calculator above as the vertical measurement labeled 'd' in the preview.

Dome

The curved surface of a hemisphere; the surface area of a hemisphere is calculated by adding the area of the dome and the area of the base.

Height

The distance measured from the origin of a hemisphere to the point on the hemisphere's surface orthogonal to the hemisphere's base; equivalent to the radius dimension.

Radius of a Hemisphere

A hemisphere's radius is the distance from the center point of a hemisphere to any point on its surface. It is the one measurement that defines the dimensions of a hemisphere. The radius is shown in the hemisphere calculator above as the horizontal measurement labeled 'r' in the preview. The hemisphere radius is also referred to as the height of the hemisphere.

Hemisphere Calculator Updates

28 August 2018

Initial version of the hemisphere calculator.

29 August 2018

Changed the the formula used by the hemisphere calculator to derive volume from area.