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# Mensuration

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by

## Kunal Jha

on 18 January 2015

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#### Transcript of Mensuration

MENSURATION
We will be covering the following figures
Square
2-D
A shape that only has two dimensions (such as width and height) and no thickness.
What is a
Two Dimensional Figure?
We will be covering the following figures:
Cube
Cuboid
Frustum
Cylinder
Cone
Sphere
3-D
A 3D figure is a figure that has height, depth, and width. Such a figure does not lie entirely in a plane. The "3" in 3D refers to the number of dimensions. The "D" stands for dimension, which is an axis in a coordinate system.
What is a
Three Dimensional Figure?
Square
A square is a regular quadrilateral.
This means that it has four equal sides and four equal angles (90-degree angles, or right angles).
It can also be defined as a rectangle in which two adjacent sides have equal length.
Area of a square
Perimeter of a Square
The Perimeter of a Square is as follows:

Perimeter= sum of the length of all sides

which gives you

P= 4s ( "s" being the side of the square )
Rectangle
A rectangle is any quadrilateral with four right angles.
It can also be defined as an
It can also be defined as a parallelogram containing a right angle.
Area of a
Rectangle
Much like the area of a square, the same formula applies to the rectangle

but there's a slight difference, as the rectangle does not have equal sides, the formula is:

Perimeter of a rectangle
The perimeter is the length of the boundry of the rectangle, so the perimeter would be:

Perimeter = 2 X the Length + 2 X the Width

the diagram on the right will help you understand this a little better
A triangle is one of the basic shapes of geometry.
It's a polygon with three corners or vertices and three sides or edges which are line segments.
A triangle with vertices A, B, and C is denoted .
There are six types of triangles
Isosceles
Scalene
Equilateral
Right angle Triangle
Obstuse Angle Triangle
Acute Angle Triangle
Area of a Triangle
The area of a Triangle has a very easy formula
which is :
Area of a = 1/2 X h X b

(where "h" is height; "b" is base)
Perimeter of a Triangle
Perimeter is the boundry of the triangle so the
perimeter would be :
Perimeter = sum of all sides
Circle
A circle is a simple closed curve which divides the plane into two regions: an interior and an exterior.
The terminologies in association with a circle are given in the box below
A circle's

Diameter
is the length of a line segment whose endpoints lie on the circle and which passes through the centre.
The
is half of the diameter
A
chord
is a line segment whose endpoints lie on the circle. (eg) the diameter is the longest chord in a circle
A
tangent

is a perpendicular line which touches any point on the circle's circumference
A
secant
is an Extended chord which runs through the circle
Trapezoid
An
arc
is a part of the circle's circumference
A
sector
is a region bounded by two radii and an arc lying between the radii
A
segment
is a region bounded by a chord and an arc lying between the chord's endpoints.
Area of a circle
The area of a circle is:

Perimeter/circumference of a circle
The circumference of the circle:
C=
Arc length
To find the length of an arc, we use this formula:
Area of Sector
Area of Arc=
A trapezoid is a quadrilateral with ONE pair of parallel sides
The parallel sides are called the bases of the trapezoid and the other two sides are called the legs or the lateral sides.
Area and perimeter of a trapezium
The area of a trapezium:

A = 1/2 X sum of parallel sides X height
Perimeter of a trapezium

P = the sum of parallel sides + length of slanting sides
Cube
Cuboid
Frustrum
Cylinder
Cone
Sphere
A cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.
²
²
²
Cube is also a cuboid.

Volume of a cuboid = length x breadth x height

All the sides of a cube are equal.

Thus, Volume of a cube = a x a x a

a
a
a
A cuboid is a solid figure bounded by six faces
Opposite sides have same area in a cuboid,

which makes the surface area formula

= b x h + b x h + l x b + l x b + l x h + l x h

which can also be written as:

total surface area = 2( bh + lb + lh )
l x b
l x b
b x h
b x h
l x h
l x h
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball.
Like a circle, which is in two dimensions, a sphere is the set of points which are all the same distance r from a given point in space.
This distance r is known as the "radius" of the sphere, and the given point is known as the center of the sphere.
The maximum straight distance through the sphere is known as the "diameter".
It passes through the center and is thus twice the radius.
A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder.
Basically a cylinder is just a circle which has been streched into a third dimension.
A frustum of a cone is the slice of a right circular cone between two planes that are perpendicular to the axis of symmetry ofthe cone. Its slant height (in the diagram) is the length of asegment from the edge of thetop to the edge of the bottom, perpendicular to both. We'll denote the radii of the top and bottom circles by r and R respectively
l = length

h = height
The volume of a cylinder :
volume = cross-sectional area x height

Since the cross - sectional area is the area of one of the circles on one side,

The volume of a cylinder is:

2
2
A cone is an -dimensional geometric shape that tapers smoothly from a base (usually flat and circular) to a point called the apex or vertex.
The axis of a cone is the straight line (if any), passing through the apex, about which the base has a rotational symmetry.
Triangle
A Few Parts of Circles
Arc Length =
n = central angle
To find the area of a sector we use
It is normally a cone or pyramid
Conical Frustum
Volume Of Conical Frustum
Volume of a Conical Frustum is Calculated by the formula
where R1, R2 are the radii of the two bases.
In geometry, the rhombic triacontahedron is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types
Rhombic triacontahedron
Surface Area of
Rhombic triacontahedron
Surface Area of Rhombic triacontahedron is calculated by the formula
Net of
Rhombic triacontahedron
S.A =
Volume of
Rhombic triacontahedron
Volume of a of Rhombic triacontahedron can be calculated by the formula
V=
where edge length of a rhombic triacontahedron is 'a'
where edge length of a rhombic triacontahedron is 'a
In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of two types.
rhombic dodecahedron
Surface Area of
rhombic dodecahedron
Surface area of rhombic dodecahedron can be calculated by the formula
S.A. =
Net of
Rhombic Dodecahedron
Volume of a
rhombic dodecahedron
Volume of Rhombic Dodecahedron can be calculated by
v=
Volume of Cylinder
The formula for finding the surface area of a cylinder is, with h as height, r as radius, and S as surface area is
Surface Area of Cylinder
where l is the lateral height of the cone
Surface area of cone can be calculated by the formula
Surface Area of Cone
Volume of Cone
The volume V of any conic solid is one third of the product of the area of the base B and the height H is
Surface area of Sphere
Surface Area of Sphere can be calculated by the formula
In 3 dimensions, the volume inside a sphere (that is, the volume of a ball) is derived to be

where r is the radius of the sphere
Volume of Sphere
Some exciting complex shapes
Rectangle

Circle

Triangle

Trapezium
Area of Square can be calculated by the formula
where 'a' is the measure of the side of the square
Parallelogram
A Parallelogram is a simple (non self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure.
Parallelogram
The area K of the parallelogram to the right (the dark area) is the total area of the rectangle less the area of the two light color triangles.
Area of Parallelogram
Day In And Day Out, We See A Variety of objects and Shapes All Around Us
many of these objects play an integral part in our lives
But have we ever really thought about these objects ?
do they have a known shape?
how much space might they occupy?
do these shapes have definite properties?
can these properties be studied mathematically ?
How can this study help us?
it is this very study
that we call
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