Sphere, Hemisphere, Spherical Shell: Surface areas and Volume

Sphere

The set of all points in space which are equidistant from a fixed point is called a sphere. A sphere is a 3-D object having both Surface area and Volume.

Formulas of Sphere 

For a sphere of radius r,

(i) Surface areas of a sphere = $4\pi {{r}^{2}}$   sq. unit

(ii) Volume of a sphere = $\frac{4}{3}\pi {{r}^{3}}$   cubic unit

Hemisphere

A Sphere when divided into two equal parts from its centre forms hemispheres.

Formula of Hemisphere

For a hemisphere with radius r,

(i) Curved Surface area of a hemisphere = $2\pi {{r}^{2}}$   sq. unit

(ii) Total Surface area of a hemisphere = Base area of hemisphere + Curved Surface area of hemisphere
                  $=2\pi {{r}^{2}}+\pi {{r}^{2}}=3\pi {{r}^{2}}$    sq. unit

(iii) Volume of a hemisphere = $\frac{1}{2}\times $   Volume of Sphere

             =$\frac{1}{2}\times \frac{4}{3}\pi {{r}^{3}}$

             =$\frac{2}{3}\pi {{r}^{3}}$   cubic unit

Spherical Shell

It is a hollow sphere with some thickness of the material with which it is made up of. For example, a tennis ball.

Formulas of Spherical Shell

 
For a Spherical shell of inner radius r and outer radius R,

(i) Outer surface area of spherical shell = $4\pi {{R}^{2}}$   sq. unit

(ii) Volume of spherical shell(material used) = Volume of Outer sphere – Volume of Inner sphere

              = $\frac{4}{3}\pi {{R}^{3}}-\frac{4}{3}\pi {{r}^{3}}$

              = $\frac{4}{3}\pi ({{R}^{3}}-{{r}^{3}})$   cu. unit


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