Mensuration Maths Notes

Mensuration Maths Notes

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List of Important Formulae:

Cuboid:

Cuboid is made up of six surfaces. Each surface is a rectangle.
Mensuration Maths Notes 1
Examples: Brick, matchbox, fish tank, etc.
If l, b, h denote the length, breadth and height of a cuboid, then

  1. lateral (vertical) surface area = 2(1 + b)h
  2. total surface area = 2 (lb + bh + hl)
  3. volume = lbh.

Mensuration Maths Notes

Cube:

Cube is made up of six surfaces. Each surface is a square. All squares are equal.
Mensuration Maths Notes 2
Examples : Ice cubes, sugar crystals, die, etc.
If l is the length of each edge of a cube, then

  1. lateral (vertical) surface area = 4l2
  2. total surface area = 6l2
  3. volume = l3

Cylinder:

Cylinder is made of three surfaces. Base surfaces are circular in shape. It also has a curved surface.
Mensuration Maths Notes 3
Examples : Solid pipe, ball pens refills, road roller, coins, etc.
If r is the base radius and h is the height of a right circular cylinder, then

  1. area of the base surface = πr2
  2. curved šurface area = 2πrh
  3. total surface area = 2πr(r + h)
  4. volume = πr2h.

Sphere:

Sphere has only one surface which is entirely curved.
Mensuration Maths Notes 4
Examples: Football, marbles, soap bubbles, lead shots, etc.
If r is the radius of a sphere, then

  1. surface area = 4πr2
  2. volume = \(\frac{4}{3}\)πr3

Mensuration Maths Notes

Hemisphere:

Hemisphere has two surfaces. The base surface is circular in shape. It also has a curved surface.
Mensuration Maths Notes 5
Examples: Bowl, broken coconut, etc.
If r is the radius of a hemisphere, then

  1. area of the base = πr2
  2. curved surface area = 2πr2
  3. total surface area = 3πr2
  4. volume = \(\frac{2}{3}\)πr3.

Right Circular Cone:

A cone has two surfaces. Base surface is circular in shape. It also has curved surface.
Mensuration Maths Notes 6
Examples: Conical tent, ice cream cone, tapered end of pencil, etc.
For a right circular cone of height h, slant height l and base radius r, we have,

  1. area of the base = πr2
  2. l2 = r2 + h2 .
  3. curved surface area = πrl
  4. total surface area = πr(r + l)
  5. volume = \(\frac{1}{3}\) πr2h.

Frustum of the Cone:

For a given cone, if we slice or cut a plane parallel to its base and remove the smaller cone, the left over part on other side of plane is called frustum of the cone. It has 3 surfaces. Two faces are circular and one is curved.
Mensuration Maths Notes 7
Examples : Bucket, water glass, etc.
If h is the height, r1, r2 are the radii of the bases (r1 > r2) and the l is the slant height of the frustum of
the cone, then

  1. l = \(\sqrt{h^{2}+\left(r_{1}-r_{2}\right)^{2}}\).
  2. curved surface area = πr(r1 + r2)
  3. total surface area = πl(r1 + r2) + πr12 + πr22
  4. volume = \(\frac{1}{3}\)πh (r12 + r22 + r1 × r2)

[Note: If we consider r2 > r1 then l = \(\sqrt{h^{2}+\left(r_{2}-r_{1}\right)^{2}}\)]

Mensuration Maths Notes

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Sector of a Circle:

In the figure, the central angle divides the circular region in two parts. The sector of a circle is the region bounded by two radii and the arc of the circle. In the figure. arc ACB and arc ADB are the minor and the major arcs respectively.
Mensuration Maths Notes 8

The region bounded by the two radii and the minor arc is called minor sector and the region bounded by the two radii and the major arc is the major sector.

In the figure, the region O-ACB is minor sector and the region O-ADB is the major sector.
Let m(arc ACB) = ∠AOB = θ

Area of the sector = \(\frac{\theta}{360}\) × πr2.
Circumference of the circle = 2πr or πd
Arc being the part of the circle, its length can be calculated.

Length ofan arc = \(\frac{\theta}{360}\) × 2πr
The relation between the length of the arc and the area of the sector is shown by the following formula :
Mensuration Maths Notes 9

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Segment of a circle:

The region bounded by a chord and its corresponding arc is called the segment of the circle.

If the corresponding arc is minor, then the segment is minor segment and if the corresponding arc is major, then the segment is major segment.
Mensuration Maths Notes 10
In the figure, the shaded region PRQ is the minor segment and region PMQ is the major segment of the circle.
Mensuration Maths Notes 11
In the figure, if we draw radii OP and OQ. we also obtain sector O-PRQ.
Thus,
Area of segment PRQ = Area of sector O-PRQ-Area of triangle OPQ.
[Also the following formula can be used to find the area of the segment:]
Area of the segment = r2\(\left[\frac{\pi \theta}{360}-\frac{\sin \theta}{2}\right]\) where r is the radius of the circle and θ is the angular measure of the arc contained by the sector.

Maths Notes

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