Science Exhibition 1000 words writeup

Jawaharlal Nehru Science, Mathematics and Environmental Exhibition Government Model Sanskriti Sen. Sec. School, Mandi Dabwali THEME :- Technology and toys SUB THEME :- MAT HEMATICS FOR US :-Name of Area in 2D and Factorization using Area Model :-Name of Dildeep Singh Insan Participant th Class :- 11

Why ?

W is the area of a circle pi times the square of the radius ? The usual definition of pi is the ratio of the circumference of a circle to its diameter, so that the circumference of a circle is pi times the diameter, or 2 pi times the radius. This model shows that a circle can be divided into concentric rings which can be unrolled to closely resemble a triangle (with height r and base 2 pi times r) of area pi times the square of the radius. By dividing the circle into more rings and unrolling, the approximation becomes better. Taking more and more rings, the triangle of area \"pi r squared\" approximates the area of the circle arbitrarily close. This gives a geometric justification that the area of a circle really is \"pi r squared\".

R Area of E C T A N G L WWhhyy ?? E Length X Breadth

Is Length×Breadth the real Area of a rectangle ? My model is totally based on Visualization of the formulas Here, I have a rectangle with Length and Breadth L and L' respectively. I have divided Breadth of rectangle to too.. small individual strings such that we can say that Length of 1 string = area occupied by the string i.e. Area of 1 string = Length of that string Area of 1 string = L Area of 'n' strings = n×L }i.e. Total Area = nx L ..................(a) L' 'n' partitions L

n is variable, It depends upon breadth ∝n is directly proportional to Breadth of 1 string. n Breadth of 1 string......(i) Breadth of 1 string is directly proportional to L' ∝Breadth of 1 string L'...........(ii) n ∝From eq. (i) and (ii) L' n = k×L' (Here, K is a constant number ) And we know that value of constant depends on experimental errors and results. Here, k=1 and hence w e can replace n with L' i.e. Breadth of rectangle Putting n=L' in eq. (a) Area of rectangle = L x L' Area of Rectangle = Length x Breadth

Why And How Area of triangle

In this model, we can Visualize that why the formula to calculate area of triangle is 0.5×base×height A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. To derive Formula, we have to play with this shape. First of all we have to draw a similar triangle inverted on the previous triangle as shown above. Cut and rearrange the newly formed parallelogram in such a way that it results in a Square.

Now, Area of Rectangle = Area of two triangles Base x Height = Area of 2 Triangles Area of 1 Triangles = Base x Height 2 =AREA OF 1 X BASE X HEIGHT 2 TRIANGLE

by Completing the Square METHORD

Completing the Square Methord Visualized In a modern Updated Version...... In elementary algebra, completing the square is a technique that is used to write a quadratic expression in a way such that it contains the perfect square and hence generally used to factorize a quadratic equation. In other words, completing the square places a perfect square trinomial inside of a quadratic expression. In this methord, generally we start by expanding the coefficients of : x^2​ (forms Square of unknown length x ), x(forms Rectangles with one unknown side), and x^​0(forms Square of any known side). Our next step is to arrange all these in such a way that all these form a Rectangle (collectively). The sides of rectangle will be the required factors.

Completing the square is used in: solving quadratic equations, deriving the quadratic formula, graphing quadratic functions, evaluating integrals in calculus, such as Gaussian integrals with a linear term in the exponent, finding Laplace transforms. and many more... In mathematics, completing the square is often applied in any computation involving quadratic polynomials

PPyytthhaaggoorraass TThheeoorreemm''ss P Vroof isualized

In this model, a square with sides (a+b) is drawn in such a way that when we join the point of each side at the point where the line is divided into two parts(a and b), it results in the formation of a square with side 'c' (say) inside a large square of side (a+b). Area of Area of [ square with large = side 'c' and triangle square with sides 'a' and 'b'] (a+b)2 = C2 + 4(1/2×a×b) a2 +b2 +2ab = c2 +2ab a2 +b2 = c2 a2 +b2 = c 2

A generalization of this theorem is the law of cosines, which allows the computation of the length of any side of any triangle, when the lengths of the other two sides and the angle between them is given. Pythagoras theorem is mentioned in the Baudhayana Sulba-sutra of India, which was written between 800 and 400 BCE. Nevertheless, the theorem came to be credited to Pythagoras.