Question

Plzz Help me guys..

✪ I am having problem in surface areas And volume.✪

Please explain the formulas that how to derive them. ( class 9)

And Please tell me how I can easily solve those questions ??


✪MATHS ARYABHATTA

Answer 1

Hello friend...

Surface area of different solids...

Surface Area of a Cube = 6 a 2

(a is the length of the side of each 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.

 

Surface Area of a Rectangular Prism = 2ab + 2bc + 2ac

 (a, b, and c are the lengths of the 3 sides)

In words, the surface area of a rectangular prism 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 (side lengths a and c) = a*c. Since there are two of them, you get 2ac. The front and back have side lengths of b and c. The area of one of them is b*c, and there are two of them, so the surface area of those two is 2bc. The left and right side have side lengths of a and b, so the surface area of one of them is a*b. Again, there are two of them, so their combined surface area is 2ab.

 Surface Area of Any Prism

 (b is the shape of the ends)

Surface Area = Lateral area + Area of two ends

(Lateral area) = (perimeter of shape b) * L

Surface Area = (perimeter of shape b) * L+ 2*(Area of shape b)

 Surface Area of a Sphere = 4 pi r 2

 (r is radius of circle)

 Surface Area of a Cylinder = 2 pi r 2 + 2 pi r h

 (h is the height of the cylinder, r is the radius of the top)

Surface Area = Areas of top and bottom +Area of the side

Surface Area = 2(Area of top) + (perimeter of top)* height

Surface Area = 2(pi r 2) + (2 pi r)* h

In words, the easiest way is to think of a can. The surface area is the areas of all the parts needed to cover the can. That's the top, the bottom, and the paper label that wraps around the middle.

You can find the area of the top (or the bottom). That's the formula for area of a circle (pi r2). Since there is both a top and a bottom, that gets multiplied by two.

The side is like the label of the can. If you peel it off and lay it flat it will be a rectangle. The area of a rectangle is the product of the two sides. One side is the height of the can, the other side is the perimeter of the circle, since the label wraps once around the can. So the area of the rectangle is (2 pi r)* h.

Add those two parts together and you have the formula for the surface area of a cylinder.

Surface Area = 2(pi r 2) + (2 pi r)* h

 
Please Don't forget the units.

These equations will give you correct answers if you keep the units straight. For example - to find the surface area of a cube with sides of 5 inches, the equation is:

Surface Area = 6*(5 inches)2

= 6*(25 square inches)

= 150 sq. inches

If you learn all these formulas then you can solve any problem related to this topic easily...

Hope this helps you...

  

Answer 2

Heya Sista and All Others !!!

<<<<FORMULAS OF SURFACE AREA and VOLUME>>>>

For me, surface areas and volume [Mensuration] is most easy. This is because everything is based on a formula. What you have to do is Just put a formula according to the Question and Then Solve. You can do the Question in just a few minutes. 

Derivation of formula's is not so simple...
But, simple derivation is given in NCERT Book before Introduction of each new Formulas.

Let us Start with Cuboid :- 

Total Surface area of Cuboid = 2 (lb+bh+lh)


A cuboid has 6 faces.
One top +One base + Two Faces {in Front and back of each other} + Two Face {one face on the Both side}

The Top and Base are Equal to each other.
They are in shape of rectangle. So their area = length X breadth = lb
So, area of two faces, i.e, base and top = 2lb.

The Front and back bases are equal.
They are also in the shape of rectangle. So their area = length X height
Area of both faces = 2lh

Area of side faces = bh
Area of two = 2bh.

Now, Whole Surface area will be = Area of all faces.
= 2lb + 2lh + 2bh
= 2(lb+lh+bh)
= 2 (lb+bh+lh).

Now, Volume of same...
Volume = Total cubic units contained in a box.
So, Volume of cuboid = Length X Breadth X Height
= lbh.

Coming to cube....
As a cube is a also a cuboid. So, we can say that 
TSA of Cube = 2 (lb+bh+lh)
but in cube, L = B = H = a.
So, TSA of cube = 2 (aXa+aXa+aXa)
= 2 (a² + a² + a²)
= 2 (3a²)
= 6a².

If you have to find LSA, 
LSA is the area of four walls,, So u can remove area of base and to from TSA to get LSA.
eg, TSA - Area of Base and Top = LSA
2lb + 2bh + 2lh - 2lb
= 2bh+2lh = 2h (l+b).

<<<<CYLINDER>>>>

A cylinder is a round shaped with some height.
If we cut a cylinder anywhere from the middle, we will get a rectangle with one circle on both sides.
whose height will be same as of the cylinder.
So, breadth will be = h
length = circumference of cylinder = 2πr
Hence, CSA = lb = 2πrh

For TSA, two circles will also be added from top and bottom.
= CSA + 2(πr²)
= 2πrh + 2πr²
= 2πr (r+h)
VOLUME = Height X area of base
= h X πr²
= πr²h.

Similarly, we can take out other formulas.
Hope It helps You.