I. Results on Triangles:

1. Sum of the angles of  a triangle is 180o

2. The sum of any two sides of a triangle is greater than the third side.

3. Pythagoras Theorem : In a right - angled triangle,


4. The line joining the mid-point of a side of a triangle to the opposite vertex is called the median.

5. The point where the three medians of a triangle meet, is called Centroid. The centroid divides each of the medians in the ratio 2 : 1.

6. In an Isosceles triangle, the altitude from the vertex bisects the base.

7. The median of a triangle divides it into two triangles of the same area.

8. The area of the triangle formed by joining the mid-points of the sides of a given triangle is one-fourth of the area of the given triangle.


II.Results on Quadrilaterals :

1. The diagonals of a parallelogram bisect each other

2. Each diagonal of a parallelogram divides it into two triangles of the same area.

3. The diagonals of a rectangle are equal and bisect each other.

4. The diagonals of a square are equal and bisect each other at right angles

5. The diagonals of a rhombus are unequal and bisect each other at right angles

6. A parallelogram and a rectangle on the same base and between the same parallels are equal in area.

7. Of all the parallelogram of given sides, the parallelogram which is a rectangle has the greatest area.




1. Area of a rectangle = (length x Breadth)

Length =AreaBreadth  and  Breadth=AreaLength

2. Perimeter of a rectangle = 2( length + Breadth)



II. Area of square = side2=12diagonal2 


III. Area of 4 walls of a room = 2(Length + Breadth) x Height




1. Area of a triangle =12×base×height

2. Area of a triangle = s(s-a)(s-b)(s-c), where a, b, c are the sides of the triangle and s=12a+b+c

3. Area of an equilateral triangle =34×side2

4. Radius of incircle of an equilateral triangle of side a=a23

5. Radius of circumcircle of an equilateral triangle of side a=a3

6. Radius of incircle of a triangle of area  and semi-perimeter s=s




1. Area of a parallelogram = (Base x Height)

2. Area of a rhombus = 12×Product of diagonals

3. Area of a trapezium = 12×(sum of parallel sides)×distance between them




1. Area of a cicle = πR2, where R is the radius.

2. Circumference of a circle = 2πR.

3. Length of an arc = 2πRθ360, where θ is the central angle.

4. Area of a sector = 12arc×R=πR2θ360 



1. Area of a semi-circle = πR22

2. Circumference of a semi - circle = πR


If each side of a square is increased by 25%, find the percentage change in its area?

A) 65.25 B) 56.25
C) 65 D) 56
Answer & Explanation Answer: B) 56.25


let each side of the square be a , then area = a2 

As given that The side is increased by 25%, then 

New side = 125a/100 = 5a/4 


New area = 5a42  


Increased area= 25a216-a2 


Increase %=9a2/16a2*100  % = 56.25%

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941 137176

If the length of the diagonal of a square is 20cm,then its perimeter must be

a. 402cm     b. 302cm    c. 10cm    d. 152cm

A) a B) b
C) c D) d
Answer & Explanation Answer: A) a


We know that d=√2s 

Given diagonal = 20 cm 

=> s = 20/2 cm

Therefore, perimeter of the square is 4s = 4 x 20/2 = 402   cm.

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72 90440

If the diagonal of a rectangle is 17cm long and its perimeter is 46 cm. Find the area of the rectangle.

A) 110 B) 120
C) 130 D) 140
Answer & Explanation Answer: B) 120


let length = x and breadth = y then  

2(x+y) = 46         =>   x+y = 23  

x²+y² = 17² = 289  

now (x+y)² = 23² 

=> x²+y²+2xy= 529 

=> 289+ 2xy = 529

=> xy = 120  

area = xy = 120 sq.cm

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278 76332

A rectangular park 60 m long and 40 m wide has two concrete crossroads running in the middle of the park and rest of the park has been used as a lawn. If the area of the lawn is 2109 sq. m, then what is the width of the road?

A) 1 B) 2
C) 3 D) 4
Answer & Explanation Answer: C) 3


Area of the park = (60 x 40) = 2400m2

Area of the lawn = 2109m2 

Area of the crossroads = (2400 - 2109) = 291m2 

Let the width of the road be x metres. Then,



  (x - 97)(x - 3) = 0
   x = 3.

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81 72413

 The percentage increase in the area of a rectangle, if each of its sides is increased by 20%

A) 22% B) 33%
C) 44% D) 55%
Answer & Explanation Answer: C) 44%


Let original length = x metres and original breadth = y metres.

Original area = xy sq.m 

Increased length  = 120100 and Increased breadth = 120100 


New area = 120100x*120100y=3625xy m2 


The difference between the Original area  and New area  is:  



1125xy Increase % =1125xyxy*100= 44%

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107 67031

A man walked diagonally across a square lot. Approximately, what was the percent saved by not walking along the edges?

A) 30 B) 40
C) 50 D) 60
Answer & Explanation Answer: A) 30


Let the side of the square(ABCD) be x meters.


Then, AB + BC = 2x metres.


AC = 2x = (1.41x) m.


Saving on 2x metres = (0.59x) m.


Saving % =0.59x2x*100 = 30% (approx)

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95 56689

If the radius of a circle is decreased by 50%, find the percentage decrease in its area.

A) 55% B) 65%
C) 75% D) 85%
Answer & Explanation Answer: C) 75%


Let original radius = R. 


New radius = 50100R50100R 


Original area =R2  and new area = πR2




Decrease in area = πR22=πR24 = 75%

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146 56343

The area of the largest circle that can be drawn inside a rectangle with sides 18cm by 14cm is

A) 49 B) 154
C) 378 D) 1078
Answer & Explanation Answer: B) 154


The diameter is equal to the shortest side of the rectangle.

 So radius= 14/2 = 7cm. 

Therefore,  area of circle =πr2=227×49=154cm2

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85 52377