Surface Areas and Volumes

Introduction:
Mensuration is a topic in Geometry which is a branch of mathematics. Mensuration
deals with the study of different geometrical shapes like length, areas and Volume both 2D
and 3D. In the broadest sense, it is all about the process of measurement.

A 2D shape is a shape that is bounded by three or more straight lines or a closed

circular line in a plane. These shapes have no depth or height; they have two dimensions-
length and breadth and are therefore called 2D figures or shapes. For 2D shapes, we measure
area (A) and perimeter (P).

A 3D shape is a shape that is bounded by a number of surfaces or planes. These are

also referred to as solid shapes. These shapes have height or depth unlike 2D shapes, they
have three dimensions - length, breadth and height depth and are therefore called 3D figures.
30 shapes are actually made up of a number of 20 shapes. Also, known as solid shapes, for
3D shapes we measure Volume (V), Curved Surface Area (CSA), Lateral Surface Area
(LSA) and Total Surface Area (TSA).

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Important Terms in Mensuration:
Before moving ahead to the list of important mensuration formulas, we need to
discuss some important terms that constitutes these mensuration formulae.
Area (A) - The surface occupied by a given closed shape is called its area. It is represented by
the alphabet A and is measured in unit square. Example: .
Perimeter (P) - The length of the boundary of a figure is called its perimeter. In other words,
it is the continuous line along the periphery of the closed figure. It is represented by the
alphabet P and in measures in cm/m.
Volume (V) - The space that is contained in a three-dimensional shape is called its volume. In
other words, it is actually the space that is enclosed in a 3D figure. It is represented by the
alphabet V and is measured in .
Curved Surface Area (CSA) - In solid shapes where there is a curved surface, like a sphere or
cylinder, the total area of these curved surfaces is the Curved Surface Area. The acronym for
this is CSA and it is measured in or .
Lateral Surface Area (LSA) - The total area of all the lateral surfaces of a given figure is
called its Lateral Surface Area. Lateral Surfaces are those surfaces that surround the object.
The acronym for this is LSA and it is measured in or .
Total Surface Area (TSA) - The sum of the total area of all the surfaces in a closed shape is
called its Total Surface Area. For example, in a cuboid when we add the area of all the six
surfaces we get its Total Surface Area. The acronym for this is TSA and it is measured in in
or .

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Square Unit - One square unit is actually the area occupied by a square of side
one unit. When we measure the area of any surface we refer to this square of side one unit
and how many such units can fit in the given figure. It is expressed as or , depending
on the unit in which the area is being measure.
Cube Unit ( ) - One cubic unit is the volume occupied by a cube of side one unit.
When we measure the volume of any figure we actually refer to this cube of side one unit and
how many such unit cubes can fit in the given closed shape. It is written in or ,
depending on the unit that is being used to measure.

Mensuration Formulas:
Every 2D and 3D figure has a list of mensuration formulas that establish a
relationship amongst the different parameters. Let's discuss the mensuration formulas of
some shapes.

List of Mensuration Formulas:

Square: Rectangle:

Area = sq. units Area = sq. units

Perimeter = 4a units Perimeter = 2(l+b) units

Diagonal = √2a units Diagonal =√ units

Scalene Triangle:

Area =√ sq. units

Perimeter = (a+b+c) units

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Equilateral Triangle:

Area = sq. units

Perimeter = 3a units

Isosceles Triangle:

Area = √ sq. units

Perimeter = 2a+b units

[b=base length; a= equal side length]

Right Angled Triangle:

Area = sq. units

Perimeter = units

Hypotenuse =√ units

Circle:

=2r uits
Diameter, D

Area = sq. units

Circumference = units

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Cube:

LSA = sq. units

TSA = sq. units

Volume = cubic units

Diagonal = a√3 units

Cuboid:

LSA = sq. units

TSA = sq. units

Volume = cubic units

Diagonal =√ units

Sphere:

Volume = cubic units

Surface Area = sq. units

If R and r are the external and internal radii of a spherical shell, then
its volume = [ ] cubic units

Hemisphere:

Volume = cubic units

Total Surface Area =3 sq. units

Curved Surface Area =2 sq. units

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Cylinder:

Volume = cubic units

Total Surface Area =2 +2 sq. units

Curved Surface Area =2 sq. units

Cone:

Volume = cubic units

Total Surface Area = sq. units

Curved Surface Area = sq. units

Slant Height of cone, l = units

Some other Formulae:

 Area of Pathway running across the middle of a rectangle = w(l+b-w)
 Perimeter of Pathway around a rectangle field = 2(l+b+4w)
 Ares of Pathway around a rectangle field = 2w(l+b+2w)
 Perimeter of Pathway inside a rectangle field = 2(l+b-4w)
 Area of Pathway inside a rectangle field = 2w(1+b-2w)
 Area of four walls = 2h(l+b)

Conclusion:
Mensuration is a very important topic when it comes to the geometry of the universe.
Archimedes is one of the most famous Greek mathematicians who contributed significantly
in geometry regarding the area of plane figures and areas as well as volumes of curved
surfaces. By definition, mensuration refers to the part of geometry concerned with
ascertaining lengths, areas, and volumes. Hence, it is easy to see why mensuration is
instrumental and plays a big part in real-world applications.

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Circles

Introduction:
A circle is a simple shape of Euclidean geometry consisting of those points in a
plane which are the same distance from a given point called the centre. The common distance
of the points of a circle from its centre is called its radius.

Circles are simple closed curves which divide the plane into two regions, an interior
and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to
either the boundary of the figure (known as the perimeter) or to the whole figure including its
interior. However, in strict technical usage, "circle” refers to the perimeter while the interior
of the circle is called a disk. The circumference of o circle is the perimeter of the circle
(especially when referring to its length).

A circle is a special ellipse in which the two foci are coincident. Circles are conic
sections attained when o right circular cone is intersected with a plane perpendicular to the
axis of the cone.

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Further terminology:
Diameter: The diameter of a circle is the length of a line segment whose endpoints lie
on the circle and which passes through the centre of the circle. This is the largest distance
between any two points on the circle. The diameter of a circle is twice its radius.
Radius: The term ‘radius’ can also refer to a line segment from the centre of a circle
to its perimeter, and similarly the term "diameter" con refer to a line segment between two
points on the perimeter which passes through the centre. In this sense, the midpoint of a
diameter is the centre and so it is composed of two radii.
Chord: A chord of a circle is a line segment whose two endpoints lie on the circle.
The diameter, passing through the circle's centre, is the largest chord in a circle.
Tangent: A tangent to a circle is a straight line that touches the circle at a single
point. A secant is an extended chord a straight line cutting the circle at two points.
Arc: An arc of a circle is any connected part of the circle's circumference. A sector is
a region bounded by two radii and an arc lying between the radii and a segment is a region
bounded by a chord and an arc lying between the chord's endpoints.

Length of circumference:
The ratio of a circle's circumference to its diameter is π (pi), a constant that takes the
same value (approximately 3.1416) for all circles. Thus the length of the circumference (c) is
related to the radius (r) by
or equivalently to the diameter (d) by .

Area enclosed:

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The area enclosed by a circle is π multiplied by the radius squared: .
The circle is the plane curve enclosing the maximum area for a given arc length. This relates
the circle to a problem in the calculus of variations, namely the isoperimetric inequality.
Properties:
 The circle is the shape with the largest area for a given length of perimeter. (See
Isoperimetric inequality).
 The circle is a highly symmetric shape: every line through the center forms a line of
reflection symmetry and it has rotational symmetry around the center for every angle.
Its symmetry group is the orthogonal group (2R). The group of rotations alone is the
circle group T.
 All circles are similar.
o A circle's circumference and radius are proportional. T
o The area enclosed and the square of its radius are proportional.
o The constants of proportionality are 2π and π, respectively.
 The circle centred at the origin with radius 1 is called the unit circle.
o Thought of as a great circle of the unit sphere, it becomes the Riemannian circle.
 Through any three points, not all on the same line, there lies a unique circle In
Cartesian coordinates, it is possible to give explicit formulae for the coordinates of the
center of the circle and the radius in terms of the coordinates of the three given points.
See circumcircle.

Chord:
 Chords are equidistant from the center of a circle if and only if they are equal in
length.
 The perpendicular bisector of a chord passes through the center of a circle, equivalent
statements stemming from the uniqueness of the perpendicular bisector.
o A perpendicular line from the center of a circle bisects the chord.
o The line segment (circular segment) through the center bisecting o chard is
perpendicular to the chord.
 If a central angle and on inscribed angle of a circle are subtended by the same chord
and on the same side of the chord, then the central angle is twice the inscribed angle.

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  If two angles are inscribed on the same chord and on the same side of the chord, then
they are equal.
 If two angles are inscribed on the same chord and on opposite sides of the chord, then
they are supplemental.
o For a cyclic quadrilateral, the exterior angle is equal to the interior opposite angle
 An inscribed angle subtended by a diameter is a right angle.
 The diameter is the longest chord of the circle.

Tangent:
 The line drawn perpendicular to a radius through the end point of the radius is a
tangent to the circle.
 A line drown perpendicular to a tangent through the point of contact with a circle
passes through the center of the circle.
 Two tangents can always be drawn to a circle from any point outside the circle, and
these tangents are equal in length.

Parts of Circles:

 Centre: It is in the centre of the circle and the distance from this point to any other
point on the circumference is the same.
 Radius: The distance from the centre to any point on the circle is called the radius. A
diameter is twice the distance of a radius.
 Circumference: The distance around a circle is its circumference, is also the
perimeter of the circle.
 Arc: An arc is a part of the circumference of a circle. The longer arc is called the
major arc while the shorter one is called the minor arc.
 Chord: A chord is a straight line joining two points on the circumference. The
longest chord in a called a diameter. The diameter passed through the centre.
 Sector: A sector is a region enclosed by two radii and an art, Refer to the figure
given. ROS is called the angle subtended by the an: RS at the centre O. The larger
sector is called the major sector while the smaller one, a minor sector.

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  Segment: A segment of a circle is the region enclosed by a chord and an arc of the
circle. The larger segment is the major segment while the smaller one is the minor
segment.
 Secant: A secant is a straight line cutting at two distinct point.
 Tangent: If a straight fine and a circle have only one point of contact, then that line is
called a tangent. A tangent is always perpendicular to the to the radius drawn to the
point of contact. This property is abbreviated as tan nad.

Theorems of Circle:

1. If 2 arcs of a circle are congruent, then corresponding chords are equal.

2. If 2 chords of a circle are equal then their corresponding arcs are congruent.

3. The perpendicular from the centre of a circle to a chord bisects the chord.

4. The line joining the centre of a circle to the mid point of a chord is perpendicular to the
chord.

5. There is only one and only circle passing through three non-collinear points.

6. Equal chords are equidistant from centre.

7. Chords of a circle which are equidistant from the centre are equal.

8. Equal chords of congruent circles are equidistant from the corresponding centers

9. Chords f congruent circles which are equidistant from the corresponding centres are equal.

10. Equal chords of a circle subtend equal angles at the centre.

11. If angles subtended by 2 chords of a circle at the centre are equal, the chords are equal.

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12. Equal chords of congruent circles subtend equal angles at the centre.

13. If the angles subtended by 2 chords of congruent circles at the corresponding centers are
equal then the chords are equal.

14. The angle subtended by an arc of a circle at the centre is double the angle subtended by it at
any point on the remaining part of the circle.

15. Angles in the same segment of a circle are equal.

16. The angle in a semi-circle is a right angle.

17. A tangent to a circle is perpendicular to the radius at the point of contact.

18. The lengths of tangents drawn from an external point to a circle are equal.

19. If 2 tangents are drawn from an external point then, they subtend equal angles at the centre.

20. If a chord is drown through the point of contact of a tangent to a circle, then the angles
which the chord makes with the given Tangent are equal to the angles formed in the
corresponding alternate segment.

Secant-secant theorem:

B
A
C

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  The chord theorem states that “If two chords, CD and EB, intersect at A, then
”.
 If a tangent from an external point D meets the circle at C and a secant from the
external point D meets the circle at G and E respectively, then
(Tangent-secant theorem.)
 If two secants, DG and DE, also cut the circle at Hand F respectively, then
DH×DG=DF×DE (Corollary of the tangent-secant theorem)
 The angle between a tangent and chord is equal to the subtended angle on the opposite
side of the chord (Tangent chord property).
 If the angle subtended by the chord at the center is 90 degrees then , where
l is the length of the chord and r is the radius of the circle.
 If two secants are inscribed in the circle as shown at right, then the measurement of
angle A is equal to one half the difference of the measurements of the enclosed arcs
(DE and BC). This is the secant-secant theorem.

Length of Arc & Area of Sector:

Concentric Circles:
Definition: Circles that have their centers at the same point.

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Concentric circles are simply circles that all have the same center. They fit inside each other
and are the same distance apart all the way around. In the figure above, resize either circle by
dragging an orange dot and see that they both always have a common center point.

Use of circle:
 Circle-shaped bottle cover
 Circle design on jewellery
 Coin
 CD
 Bicycle wheels
 Circular wave of water

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Probability

Introduction:
Probability is a concept which numerically measures the degree of certainty of the
occurrence of events. Before defining probability, we shall define certain concepts used there.

 Experiment: An operation which can produce more well- defined outcomes is called
an experiment.
 Random Experiment: An experiment in which all possible outcomes are known, and
the exact outcome cannot be predicted in advance, is called a random experiment.
 Trial: Means 'performing a random experiment'.

Examples: (i) Tossing a fair coin

(ii) Drawing a card from a pack of well-shuffled cards.
These are all examples of a random experiment.

Some details about these experiments:

1. Tossing a coin - When we throw a coin, either a head (H) or a tail (T) appears on the upper
face.
2. Drawing a card from a well-shuffled deck of 52 cards.
(i) It has 13 cards each of four suits, namely spades, clubs, hearts and diamonds.
(a) Cards of spades and clubs are black cards.
(b) Cards of hearts and diamonds are red cards.
(ii) Kings, queens and jacks (or knaves) are known as face cards. Thus, there are in all 12
face cards.

Looking at all possible outcomes in various experiments:

(i) When we toss a coin, we get either a head (H) or a tail (T). Thus, all possible outcomes are
H, T:
(ii) Suppose two coins are tossed simultaneously. Then, all possible outcomes are HH, HT,
TH, TT (HH means head on first coin and head on second coin. HT means head on first coin
and tail on second coin etc.)

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(iii) In drawing a card from a well-shuffled deck of 52 cards, total number of possible
outcomes is 52.

 Event: The collection of all or some of the possible outcomes is called an event.
Examples: (i) In throwing a coin. H is the event of getting a head.
(ii) Suppose we throw two coins simultaneously and let E be the event of getting at
least one head. Then, E contains HT, TH, HH.

 Equally likely events: A given number of events are said to be equally likely if none
of them is expected to occur in preference to the others.
For example, if we roll an unbiased die, each number is equally likely to occur. If,
however, a die is so formed that a particular face occurs most often then the die is
biased. In this case, the outcomes are not equally likely to happen.

Probability of occurrence of an event:

Probability of occurrence of an event E. denoted by P (E) is defined as:

 Sure event: It is evident that in a single toss of die, we will always get a number less
than 7
So, getting a number less than 7 is a sure event.
P (getting a number less than 7) = 6/6 = 1.
Thus, the probability of a sure event is 1.

 Impossible event: In a single toss of a die, what is the probability of getting a

number 8?
We know that in tossing a die, 8 will never come up.
So, getting 8 is an impossible event.
P (getting 8 in a single throw of a die) = 0/60 = 0.
Thus, the probability of an impossible event is zero.

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  Complementary event: Let E be an event and (not E) be an event which occurs only
when E does not occur. The event (not E) is called the complementary event of E.
Clearly, P(E) + P(not E) = 1
P(E)=1-P(not E).

Solved examples:

Example-1: A coin is tossed once, what is the probability of getting a head?

Solution: When a coin is tossed once, all possible outcomes are H and T.
Total number of possible outcomes = 2.
The favourable outcome is H.
Number of favourable outcomes = 1.

Example-2: A die is thrown once. What is the probability of getting a prime number?
Solution: In a single throw of a die, all possible outcomes are 1,2,3,4,5,6.
Total number of possible outcomes = 6.
Let E be the event of getting a prime number.
Then, the favourable outcomes are 2,3,5 ; Number of favourable outcomes = 3.

Example-3: A bag contains 5 red balls and some blue balls. If the probability of drawing a
blue ball from the bag is thrice that of a red ball, find the number of blue balls in the bag.
Solution: Let the number of blue balls in the bag be x. Then, total number of balls = (5 + x).
Given: P(a blue ball) = 3 x P (a red ball)

Hence, the number of blue balls in the bag is 15.

Example-4: One card is drawn at random from a well-shuffled pack of 52 cards. What is the
probability that the card drawn is either a red card or a king?

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Solution: Total number of all possible outcomes = 52.
Let E be the event of getting a red card or a king. There are 26 red cards (including 2 kings)
and there are 2 more kings.
So, the number of favourable outcomes = (26+2) = 28.

Conclusion:
To conclude I should say that each student should develop interest or rather equal
interest in all topics as to serve good marks in Math's Project. A student should concentrate
more in which he/she is unable to do or in which they are lacking interest.

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 Bibliography

1. www.google.com

2. www.scribd.com

3. www.wikipedia.com

4. www.wolfram.com

5. www.mathsopenref.com

6. www.cut-the-knot.com

7. www.library.thinkquest.com

8. www.adhyapak.com

9. Online Teacher Consulted

10. Class X R. D. Sharma

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