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STUDY MATERIAL
SUB: MATHS
Topic: Mensuration 4th JUNE, 2024
Perimeter The perimeter of a plane figure is the length of its boundary.
In case of a triangle or a polygon, the perimeter is the sum of the lengths of its sides.
The unit of perimeter is same as the unit of length.

Area The area of a plane figure is the measure of the surface enclosed by its boundary.
The area of a triangle or a polygon is the measure of the surface enclosed by its sides.
Area is measured in square units such as square centimetres and square metres, written as cm2 and m2
respectively.

PERIMETER AND AREA OF TRIANGLES (FORMULAE)

1. (i) Area of a triangle
ar(∆ABC) . / sq units.
(ii) Heron’s formula

Heron’s formula, is credited to Heron of Alexandria for finding the area of a triangle in terms of the
lengths of its sides.

Let a, b, c be the sides of a ∆ABC. Then,

s ( ) is called its semiperimeter.
ar(∆ABC) √ ( )( )( )

2. In a right ∆ABC, let B = 900. Then,

ar(∆ABC) . / sq units.

3. In an equilateral triangle of side a, we have

√
(i) Height . / units.
√
(ii) Area . / units.
(iii) Perimeter = 3a units.

4. For an isosceles ∆ABC in which AB = AC = a and BC = b, we have

√
(i) Height units
(ii) Area . √ / sq units
(iii) Perimeter = (2a + b) units
 Perimeter and Area of Quadrilaterals (Formulae)
1. Area and perimeter of a rectangle
(i) Area = (length breadth) sq units
(ii) Length and Breadth
(iii) Perimeter = 2(l + b) units
(iv) Diagonal √ units

2. Area of 4 walls of a room = [2(l + b) h)] sq unit

3. Area and perimeter of a square:

(i) Area = sq units
(ii) Area = { ( ) } sq units
(iii) Perimeter = 4a units.
(iv) Diagonal = √ units

4. (i) Area of a parallelogram = {(base) (height)} sq units.

(ii) Area of a rhombus = . / sq units

5. Area of a trapezium
( ) ( )

6. Area of a quadrilateral:
(i) Let ABCD be a quadrilateral with diagonal AC. Let BL AC and DM AC. Let BL = h1 and
DM = h2. ar(quad. ABCD) { ( ) } sq units

(ii) When diagonals of a quadrilateral are perpendicular to each other.

ar(quad. ABCD) . / sq units
 Area of Circle, Sector and Segment
Circle: The set of points which are at a constant distance of r units from a fixed point O is called a
circle with centre O and radius = r units. The circle is denoted by C(O, r)
In other words, a circle is the locus of a point which moves in such a way that its distance from a
fixed point O remains constant at r units.
The fixed point O is called the centre and the constant distance r units is called its radius.

Circumference: The perimeter (or length of boundary) of a circle is called its circumference.
Radius: A line segment joining the centre of a circle and a point on the circle is called a radius of
the circle.
Plural of radius is radii. In the given figure OA, OB, OC are three radii of the circle.
Chord: A line segment joining any two points on a circle is called a chord of the circle.
In the given figure, PQ, RS and AOB are three chords of a circle with centre O.
Diameter A chord of a circle passing through its centre is called a diameter of the circle.
Diameter is the longest chord of a circle. In the above figure, AOB is a diameter.
Diameter = 2 radius

Secant: A line which intersects a circle at two points is called a secant of the circle.
In the given figure, line l is a secant of the circle with centre O.
ARC (Arc): A continuous piece of a circle is called an arc of the circle.
In the given figure, AB is an arc of a circle, with centre O, denoted by ̂ The remaining part of
the circle, shown by the dotted lines, represents ̂ .

Central Angle: An angle subtended by an arc at the centre of a circle is called its central angle.
In the given figure of a circle with centre O, central angle of ̂ AOB = .
If then ̂ is called the minor arc and ̂ is called the major arc.

Semicircle: A diameter divides a circle into two equal arcs. Each of these two arcs is called a
semicircle.
In the given figure of a circle with centre O, ̂ and ̂ are semicircles.
An arc whose length is less than the arc of a semicircle is called a minor arc. An arc whose length
is more than the arc of a semicircle is called a major arc.

Segment: A segment of a circle is the region bounded by an arc and a chord, including the arc and
the chord.
The segment containing the minor arc is called a minor segment, while the segment containing the
major arc is the major segment.
The centre of the circle lies in the major segment.
Sector of a Circle: The region enclosed by an arc of a circle and its two bounding radii is called a
sector of the circle.
In the given figure, OACBO is a sector of the circle with centre O.
If arc AB is a minor arc then OACBO is called the minor sector of the circle.
The remaining part of the circle is called the major sector of the circle.
Quadrant: One-fourth of a circular disc is called a quadrant. The central angle of a quadrant is 900.
 Formulae on area of Circle, Sector and Segment
1. For a circle of radius r, we have
(i) Circumference
(ii) Area
2. For a semicircle of radius r, we have
(i) Perimeter ( )
(ii) Area
3. For a ring having outer radius = R and inner radius = r, we have
Area of the ring ( )
4. Let an arc ACB make an angle at the centre of a circle of radius r. Then, we have
(i) Length of minor arc ACB
Length of major arc BDA . /
(ii) Area of minor sector OACBO . /
Area of major sector OADBO . /
(iii) Area of minor segment ACBA . /.
Area of major segment BDAB , ( )-
(iv) Perimeter of sector OACBO . /
5. For Rotation of the Hands of a Clock:
(i) Angle described by minute hand in 60 minutes = 3600
(ii) Angle described by hour in 12 hours = 3600
6. For Rotating Wheels:
(i) Distance moved by a wheel in 1 rotation = its circumference
(ii) Number of rotations in 1 minute
7. Touching Circles:
(i) When two circles touch internally [see fig (i)], then distance between their centres = difference of their radii.
(ii) When two circles touch externally [see fig. (ii)], then distance between their centres = sum of their radii.

Fig. (i) Fig. (ii)

For a cuboid of length = L, breadth = b & height = h, we have volume = (l b h) cubic units.
Total surface area = 2 (lb + bh + lh) Sq. units
Lateral Surface area = [2(l + b) h] Sq. units
Diagonal = √ units
For a cube having each edge units, we have
Volume = a3 cubic units
Total surface area = 6a2 Sq. units.
Lateral Surface area = 4a2 Sq. units
Diagonal = √ a units

For a cylinder of base radius = r & height (or length) = h, we have

Volume = cu. units
Curved surface area = 2 Sq. units
Total surface area = ( ) Sq. units
= ( )

Consider a hollow cylinder having external radius = R, internal radius = r & height = h.
Volume of material = External volume – Internal volm
=
= ( ) Cu. units
Curved surface area of hollow Cylinder
= External Surface area + Internal surface area
=( )
=2 ( ) Sq. units

Total surface area of hollow cylinder = Curved surface area + area of base rings
= *( ) ( )+ Sq. units
=* ( ) ( )( )+
= ( )( ) Sq, units
Consider a cone in which base radius = r, height = h & slant height, L = √

Volm of cone = cu. units.

Curved surface area of the cone = √ Sq. units

T.S.A of Cone = Curved surface area + area of base
= ( ) Sq. units

Or a sphere of radius r, we have

Volm of sphere = Cu. units
Surface area of sphere = Sq. units

For a hemisphere of radius r, we have

Volm = cu.units
Curved surface area = sq. units
T.S.A = sq. units

Consider a spherical shell having external radius = R & internal radius = c

Volm of material = external volm – Internal volm
=. / Cu. units
= ( ) Cu. units.