Surface Areas and Volumes - Surface Area of a Sphere and Hemisphere

A sphere is a three-dimensional solid where every point on its surface is at a constant distance, known as the radius ($r$), from a fixed point called the center. Visually, it resembles a perfectly symmetrical round ball, like a marble or a planet.

The surface area of a sphere is the total area covered by its exterior boundary. An interesting visual way to understand this is that the surface area of a sphere is exactly equal to the area of four circles having the same radius ($r$).

A hemisphere is formed when a sphere is divided into two equal halves by a plane passing through its center. Visually, it looks like a bowl or a dome with a flat circular base and a curved top.

The Curved Surface Area (CSA) of a hemisphere refers only to the area of the rounded, outer part. Since a hemisphere is half of a sphere, its CSA is exactly half of the sphere's total surface area.

The Total Surface Area (TSA) of a solid hemisphere includes both the curved surface and the flat circular base. To visualize this, think of a solid wooden bowl; the TSA would include the outer rounded wood and the flat circular rim/top.

The surface area of these shapes depends solely on the radius ($r$). Because the formula involves $r^2$, if you double the radius of a sphere, its surface area increases by a factor of four ($2^2 = 4$).

Surface area is always measured in square units, such as $cm^2$ or $m^2$. When solving problems, ensure that the radius and the resulting area use consistent units.