VOLUME OF A SPHERE

A Sphere is a perfectly round geometrical and circular object in three-dimensional space that resembles the shape of a completely round ball. Like

a circle, which, in geometric contexts, is in two dimensions, a sphere is defined mathematically as the set of points that are all the same

distance r from a given point in three-dimensional space.

In 3 dimensions, the volume inside a sphere (that is, the volume of a ball) is derived to be = 4/3 * π * r^3

where r is the radius of the sphere and π is the constant pi. Archimedes first derived this formula.

SURFACE AREA OF A RIGHT CIRCULAR CONE

SURFACE AREA OF A CUBOIDSurface Area of a Cuboid = 2(lb + bh + hl)where l, b, h are respectively the three edges of the cuboid.

In words, the surface area of a cuboid is the area of the six rectangles that cover it. But we don't have to figure out all six because we know that the top and bottom are the same, the front and back are the same, and the left and right sides are the same. The area of the top and bottom surface = 2*a*c, since there are two of them and we get 2ac. Similarly, the area of the front and back surface = 2bc and the area of the left and right surface = 2ab. When we add the area of all the surfaces, we get the total surface area of the cuboid.

EXAMPLE QUESTION A cuboidal cereal box has length 20 cm, height 30 cm and width of 8 cm. Find the surface area of the box.

SOLUTIONThe area of the front face is 20 x 30 = 600 cm2.The area of the top face is 20 x 8 = 160 cm2.

The area of the side face is 8 x 30 = 240 cm2

Surface area of a cuboid is 2(lb + bh + hl) = 2(600 + 160 + 240)= 2*1000 = 2000 cm2

SURFACE AREA OF A CUBESurface Area of a Cube = 6 a 2

where a is the edge of the cube.

In words, the surface area of a cube is the area of the six squares that cover it. The area of one of them is a*a, or a 2 . Since these are all the same, you can multiply one of them by six, so the surface area of a cube is 6 times one of the sides squared.

EXAMPLE QUESTIONS

1. Find the Surface area of a cube of side length 8 cm.Solution: SA = 6 × a2

SA = 6 × (8) 2

SA = 6 × (64)SA = 384 cm2

2. Find the total surface area of a box whose edges are all 4.5 cm long.

Solution:

SA = 6× l × l

SA = 6 × 4.5 x 4.5

SA = 121.5 cm2

SURFACE AREA OF SPHEREWhat is a sphere? Is it the same as a circle? Can you draw a circle on a paper? Yes,

you can, because a circle is a plane closed figure whose every point lies at a constant

distance (called radius) from a fixed point, which is called the center of the circle.

Now if you paste a string along a diameter of a circular disc and rotate it as you had

rotated the triangle in the previous section, you see a new solid . What

Can you guess what happens to the center of the circle, when it forms a sphere on

rotation? Of course, it becomes the center of the sphere. So, a sphere is a three

dimensional figure (solid figure), which is made up of all points in the space,

which lie at a constant distance called the radius, from a fixed point called center of sphere

Note : A sphere is like the surface of a ball. The word solid sphere is used for the

solid whose surface is a sphere does it resemble? A ball? Yes. It is called a sphere.

Surface Area of a Sphere = 4 π r2

where r is the radius of the sphere.

How many faces do you see in the surface of a sphere? There is only one, which is

curved.

Now, let us take a solid sphere, and slice it exactly ‘through

the middle’ with a plane that passes through its centre. What

happens to the sphere?

Yes, it gets divided into two equal parts (see Fig. 13.20)!

What will each half be called? It is called a hemisphere.

(Because ‘hemi’ also means ‘half’)

And what about the surface of a hemisphere? How many faces does it have?

Two! There is a curved face and a flat face (base).

The curved surface area of a hemisphere is half the surface area of the sphere, which is ½ of 4πr2