Download 1. Square : a a l b 2. Rectangle : 3. Rhombus : 4. Trapezium : h a b h

Chapter 30 - Mensuration - Two dimensional
In this chapter, we will be attending to problems on area of two dimensional figures. i.e. figures
involving two linear measurements. For example a square is a two dimensional figure involving
length and breadth, whereas a cube is a 3 dimensional figure involving length, breadth and
height. First, we will have an idea of the two dimensional figures and their respective properties.
Properties:
1. 4 equal sides and the opposite sides are parallel to each
other.
2. 4 internal angles, each measuring
.
3. If the length of one side of a square is ‘a’ units,
Area of the square = square units.
1. Square :
a√2
a
4. Length of the diagonal = a .
5. Perimeter of the square = 4a.
6. Area of a square with diagonal ‘d’ units =
a
2. Rectangle :
l2 + b 2
b
l
square units.
Properties:
1. Of the 4 sides, two sides, i.e. the lengths and the other
two sides i.e. the breadths are separately equal and
parallel to each other.
2. 4 internal angles, each measuring
.
3. If the length of one side of a rectangle is ‘l’ units and the
breadth is ‘b’ units, Area of the rectangle = l x b square
units i.e. Area of a rectangle = Length x Breadth.
4. Length of the diagonal =
.
5. Perimeter of a rectangle = 2 (Length + Breadth).
Properties:
1. All the 4 sides of a rhombus are equal.
2. Opposite sides are parallel to each other.
3. The two opposite angles are equal.
3. Rhombus :
4. Area of a rhombus = x Product of its diagonals.
5. If D1 and D2 are the two diagonals of the rhombus, then
the length of one side =
4. Trapezium :
b
h
h
a
x
.
Properties:
1. The 4 sides of a trapezium may or may not be equal.
2. Of the 4 sides, any one pair of the two opposite sides
is parallel.
3. Area of a trapezium = x Sum of parallel sides x Height
=
[ a + b].
Excel Speed Maths - Chapter 30 - Mensuration - Two dimensional
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5. Quadrilateral :
H1
Properties:
1. The 4 sides are not equal.
2. The sides are not parallel to each other.
3. Area of the quadrilateral =
d
H2
x any one diagonal x
Sum of altitudes from the other two vertices to
this diagonal = [H1+H2] .
4. Sum of angles inside a quadrilateral is
.
{The sum of angles inside a polygon of ‘n’ sides
= (n - 2) x 180}
Properties:
1. If r is the radius of the circle, its diameter, d = 2r.
2. Circumference of the circle =
.
3. Area of the circle =
.
Unless otherwise
6. Circle :
A
C
B
mentioned, we will take the value of
as
4. Length of the Arc AD of the circle =
5. Area of the Sector ACD =
D
a
a
with side ‘a’ =
Isoceles triangle
a
h
.
.
Properties :
1. A triangle has 3 sides and 3 angles.
2. The sum of angles inside a triangle is equal to
.
3. When the 3 sides of a triangle are equal it is called an
equilateral triangle. Area of an equilateral triangle
7. Triangles :
Equilateral triangle
a
.
a
. In an equilateral triangle,
each of the angles measures
.
4. A triangle with any two sides equal is called an Isoceles
triangle. The angles opposite to the two equal sides
will be equal. Area of an isoceles triangle, with the
equal sides ‘a’ units each and the third side, ‘b’ units
.Height of the triangle =
.
=
5. If all the sides of a triangle are unequal, it is referred
to as a Scalene Triangle. Area of a scalene triangle
= S(S − a)(S − b)(S − c) where ‘S’ is the semi- perimeter
Scalene triangle
a
b
of the triangle =
. Area of a triangle if
perpendicular height to any one side from an
c
opposite vertex is known =
Excel Speed Maths - Chapter 30 - Mensuration - Two dimensional
x Base x Height.
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1.
2.
3.
4.
5.
6.
7.
8.
9.
Unit test - Mensuration - Area of two dimensional figures :
If the perimeter of a rectangle and a square is equal to 80 Cms each and the difference
of their areas is 100 Sq.Cm, the sides of the rectangle are ;
(a)40 Cms , 20 Cms (b) 30 Cms , 20 Cms (c) 40 Cms , 25 Cms (d) 30 Cms , 10 Cms
The cost of carpeting a room of length 8 metres is Rs.48,000. If the breadth of the
room is 2 metres less, the cost would have been Rs.16,000 less. What is the breadth
of the room?
(a) 6 metres (b) 5 metres (c) 4.5 metres (d) 4 metres
A room of length 8 metres and breadth 6 metres is to be carpeted leaving 50 Cms bare
all around. Find the cost of carpeting the room, with a carpet 70 Cms wide at the rate
of Rs.80 per metre.
(a)Rs.6800 (b) Rs.4000 (c) Rs.5500 (d) Rs.5300
Around a rectangular park of length 200 metres and breadth 150 metres,a road of width 5
metres is constructed . Find the cost of tarring the road at the rate of Rs.250 per Sq.M?
(a)Rs.8 Lakh (b)Rs.8.5 Lakh (c) Rs.9 Lakh (d) Rs.10 Lakh
A cow is tethered to the centre of a square grass field with a rope of length 7 metres,
so that the cow just touches all the four sides of the field. What is the area of the field
left ungrazed?
(a) 42 Sq.M (b) 46 Sq.M (c) 52 Sq.M (d) 58 Sq.M
A table cloth of length 8 Ft. and breadth 6 Ft. is spread on a table. Calculate the cost of
polishing the top of the table at the rate of Rs. 80 per Sq.Ft, if ½ feet of the cloth is
hanging all around the table.
(a) Rs. 2400 (b) Rs. 2600 (c) Rs. 2800 (d) Rs. 3200
The difference between the length and the breadth of a rectangle is 23 metres. If its
perimeter is 206 metres, what is the area of the rectangle?
(a) 2580 Sq.M. (b) 2520 Sq.M. (c) 2680 Sq.M. (d) 2820 Sq.M.
If the diagonals of two squares are in the ratio 2:5, their areas will be in the ratio.
(a) 1:4 (b) 4:1 (c) 4:25 (d) 25:4
Find the ratio of the radii of a circle inscribed inside a square and of a circle inscribing
the square.
(a) 1:2 (b) 2:1 (c)
:1 (d) 1:
10. The diameters of two circles are the side of a square and the diagonal of a square.
The ratio of areas of the smaller circle and the larger circle is.
(a) 1:2 (b) 2:1 (c) 1:4 (d) 4:1
11. The radius of a circular wheel is 1.75 metres. The number of revolutions it will make
on covering 33 Kms is.
(a) 3000 (b) 2500 (c) 3500 (d) 4000
12. The area of a rectangle is 396 Sq.Cm. Had the length of the rectangle been longer
by 3 Cms, the area would have been 450 Sq.Cm. The length of the rectangle is.
(a) 18 Cms (b) 20 Cms (c) 22 Cms (d) 24 Cms
13. A rectangular field has its sides in the ratio 5:3. If the area of the field is 13500 Sq.M.,
find the cost of fencing the field at the rate of Rs.45 per metre.
(a) Rs.21,600 (b) Rs.22,400 (c) Rs.23,600 (d) Rs.24,800
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Excel Speed Maths - Chapter 30 - Mensuration - Two dimensional