FACTS  AND  FORMULAE  FOR  AREA  QUESTIONS

 

 

FUNDAMENTAL CONCEPTS :

I. Results on Triangles:

1. Sum of the angles of  a triangle is 

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.

 

IMPORTANT FORMULAE

I. 

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

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

 

 

II. Area of square = 

 

 

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

 

 

IV.

1. Area of a triangle =

2. Area of a triangle = , where a, b, c are the sides of the triangle and 

3. Area of an equilateral triangle =

4. Radius of incircle of an equilateral triangle of side 

5. Radius of circumcircle of an equilateral triangle of side 

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

 

 

V.

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

2. Area of a rhombus = 

3. Area of a trapezium = 

    

 

VI.

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

2. Circumference of a circle = .

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

4. Area of a sector = 

 

 

VII.

1. Area of a semi-circle = 

2. Circumference of a semi - circle = 

Q:

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

Explanation:

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

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

New side =  = 

New area = 

increased area= 

Increase %=  % = 56.25%

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195 45489
Q:

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%

Explanation:

Let original radius = R.

New radius = 

Original area =  and new area = 

Decrease in area =  = 75%

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53 20683
Q:

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

Explanation:

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

Then, AB + BC = 2x metres.

AC =  = (1.41x) m.

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

Saving % = = 30% (approx)

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31 16604
Q:

A parallelogram has sides 30m and 14m and one of its diagonals is 40m long. Then its area is

A) 136 B) 236
C) 336 D) 436
 
Answer & Explanation Answer: C) 336

Explanation:

let ABCD be the given parallelogram

area of parallelogram ABCD = 2 x (area of triangle ABC)

now a = 30m, b = 14m and c = 40m

Area of triangle ABC = \inline {\color{Black}\sqrt{s(s-a)(s-b)(s-c)}}

                                   = \inline {\color{Black}\sqrt{42(12)(28)(2)}}= 168sq m

area of parallelogram ABCD = 2 x 168 = 336 sq m

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26 14362
Q:

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

Explanation:

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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42 13521