A connection which could be illustrated, and could be understood by students who know the perimeter of a circle, runs as follows: Put the sphere of radius R inside a cylinder, with the cylinder just touching the equator, and cut off at the height of the top and bottom of the sphere. (A cutaway view is in the diagram.) [pic]

What is the area of the curved part of the cylinder? 2 Pi R x 2R = 4 Pi R2. This is found by slicing the cylinder surface and rolling it out as a rectangle. Now, it is NOT an accident that the cylinder surface is EXACTLY the area of the sphere. Take in small horizontal slice through the diagram. (I have colored one such slice orange.) This cuts a rectangle out of the rolled out cylinder and slightly distorted rectangle out of the sphere. (If the slice is very thin then the distortion is "slight".) In the cross-sectional view below hc is the height of the slice on the cylinder, hs is the length of the arc on the sphere cut out by the slice, r is the radius of the distorted rectangle on the sphere and R is the radius of the sphere. [pic]

The area of the orange rectangle on the cylinder is 2 Pi R hc and the area of the distorted orange rectangle on the sphere is approximately 2 Pi r hs. These two areas are approximately equal (the proof is outlined below). Since the cylinder and the sphere can be decomposed into these rectangular strips, the area of the sphere and the area of the cylinder are approximately equal. This argument is in the spirit of how the Greeks compared slices to show that areas and volumes are the same. It is not only interesting reasoning in proportion, etc., but it is a lesson in History. In the same spirit, you can compare the VOLUME of a hemi-sphere with that of the cylinder with an inverted cone removed. (I think of the hemisphere with the equator at the bottom and the cone...

...SurfaceArea Formulas
In general, the surfacearea is the sum of all the areas of all the shapes that cover the surface of the object.
Cube | Rectangular Prism | Prism | Sphere | Cylinder | Units
Note: "ab" means "a" multiplied by "b". "a2" means "a squared", which is the same as "a" times "a".
Be careful!! Units count. Use the same units for all measurements. Examples
...

...the area of a square with a side length of 4 inches....C
Find the volume of concrete needed to make a circular patio that has a radius of 24 feet and is 8 feet thick. Use 3.14 for pi. If necessary, round the answer to the nearest cubic foot....C
B
Write the equation of the circle with the given center and radius. Then graph the circle.
center: (–1, –3) radius: 6.....B
A
The volumes of 2 similar solids are 27 and 125 The surfacearea of the...

...SurfaceareaSurfacearea is the measure of how much exposed area a solid object has, expressed in square units. Mathematical description of the surfacearea is considerably more involved than the definition of arc length of a curve. For polyhedra (objects with flat polygonal faces) the surfacearea is the sum of the areas of its faces. Smooth...

...
8th Maths Practice Paper-1
1. An altitude of a triangle is 5/3 of its corresponding base. If the altitude were increased by
4cm and the base decreased by 2cm, the area of the triangle would remain the same. Find the base and altitude of the triangle.
2. Some toffees are bought at the rate of 11 for Rs10 and same numbers are bought at the
rate of 9 for Rs 10. If the whole lot is sold at one rupee per toffee, find the gain or loss percent.
3. Chandu purchased...

...3/2 power of its surfacearea. [Why is this reasonable? Note: raindrops are not
spherical, but let’s assume that they always have the same shape, no matter what their size may
be.] Suppose that the rate of reduction of the volume of a raindrop is proportional to its surfacearea. [Why is this reasonable?] Find a formula for the amount of time it takes for a virga raindrop
to evaporate completely, expressed in terms of the constants...

...Section
Question 1.
a) What number must be subtracted from 2x3 – 5x2 + 5x so that the resulting polynomial has a factor 2x – 3 ? [3]
b) D, E, F are mid points of the sides BC, CA and AB respectively of a Δ ABC. Find the ratio of the areas of Δ DEF and Δ ABC. [3]
c) A man borrowed a sum of money and agrees to pay off by paying Rs 3150 at the end of the first year and Rs 4410 at the end of the second year. If the rate of compound interest is 5%...

...Mathematics
Volume of Solids
Formulae for Volume of Solids
Cube | Cuboid | Triangular Prism | Cylinder | Cone | Pyramid | Sphere | AnyPrism |
s3 | lwh | ½bhl | Πr2h | 1/3πr2h | 1/3Ah | 4/3πr3 | Ah |
A = area of the base of the figure
s = length of a side of the figure
l = length of the figure
w = width of the figure
h = height of the figure
π = 22/7 or 3.14
1. Compute the volume of a cube with side 7cm.
Volume of cube: s3
s = 7cm
s3 = (7cm...

...Intro: SurfaceArea and Volume
Multiple Choice Identify the choice that best completes the statement or answers the question. Find the surfacearea of the space figure represented by the net. ____ 1.
12 in. 4 in.
6 in.
4 in. 4 in. 6 in.
a. 288 in.2 ____ 2.
b. 144 in.2
c. 240 in.2
d. 288 in.2
5 cm 5 cm
7 cm 8 cm 4 cm
____
a. 124 cm2 b. 110 cm2 c. 150 cm2 d. 164 cm2 3. Find the surface...

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