Short Tricks to Solve Problems on Area

FUNDAMENTAL CONCEPTS

  1. Results on Triangles:
    1. Sum of the angles of a triangle is 180°.
    2. The sum of any two sides of a triangle is greater than the third side.
    3. Pythagoras Theorem:In a right-angled triangle, (Hypotenuse)2 = (Base)2 + (Height)2.
    4. The line joining the mid-point of a side of a triangle to the positive vertex is called the median.
    5. The point where the three medians of a triangle meet, is called centroid. The centroid divided 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.
  2. Results on Quadrilaterals:
    1. The diagonals of a parallelogram bisect each other.
    2. Each diagonal of a parallelogram divides it into 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

  1. 1.   Area of a rectangle = (Length x Breadth).
           Length = Area and Breadth = Area .
    Breadth Length

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

  2. Area of a square = (side)2 = (diagonal)2.
  3. Area of 4 walls of a room = 2 (Length + Breadth) x Height.
  4. 1.   Area of a triangle =  x Base x Height.2.   Area of a triangle = s(sa)(sb)(sc)
    where abc are the sides of the triangle and s = (a + b + c).

    3.   Area of an equilateral triangle = 3 x (side)2.
    4
    4.   Radius of incircle of an equilateral triangle of side a = a .
    23
    5.   Radius of circumcircle of an equilateral triangle of side a = a .
    3
    6.   Radius of incircle of a triangle of area  and semi-perimeter r = .
    s
  5. 1.   Area of parallelogram = (Base x Height).2.   Area of a rhombus =  x (Product of diagonals).3.   Area of a trapezium =  x (sum of parallel sides) x distance between them.
  6. 1.   Area of a circle = R2, where R is the radius.2.   Circumference of a circle = 2R.
    3.   Length of an arc = 2R , where  is the central angle.
    360
    4.   Area of a sector = 1 (arc x R) = R2 .
    2 360
  7. 1.   Circumference of a semi-circle = R.
    2.   Area of semi-circle = R2 .
    2

     Area – General Questions

    1. The ratio between the length and the breadth of a rectangular park is 3 : 2. If a man cycling along the boundary of the park at the speed of 12 km/hr completes one round in 8 minutes, then the area of the park (in sq. m) is:
    A. 15360
    B. 153600
    C. 30720
    D. 307200

    Answer: Option B

    Explanation:

    Perimeter = Distance covered in 8 min. = 12000 x 8 m = 1600 m.
    60

    Let length = 3x metres and breadth = 2x metres.

    Then, 2(3x + 2x) = 1600 or x = 160.

     Length = 480 m and Breadth = 320 m.

     Area = (480 x 320) m2 = 153600 m2.

    An error 2% in excess is made while measuring the side of a square. The percentage of error in the calculated area of the square is:
    A. 2%
    B. 2.02%
    C. 4%
    D. 4.04%

    Answer: Option D

    Explanation:

    100 cm is read as 102 cm.

     A1 = (100 x 100) cm2 and A2 (102 x 102) cm2.

    (A2 – A1) = [(102)2 – (100)2]

    = (102 + 100) x (102 – 100)

    = 404 cm2.

     Percentage error = 404 x 100 % = 4.04%
    100 x 100

     

     

    The ratio between the perimeter and the breadth of a rectangle is 5 : 1. If the area of the rectangle is 216 sq. cm, what is the length of the rectangle?

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

    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 towel, when bleached, was found to have lost 20% of its length and 10% of its breadth. The percentage of decrease in area is:

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

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