MENSURATION
Area of Triangle
⊗Area of the Area = ¹/₂base*height
⊗If a , b and c be the the sides of the
traingle ,
then perimeter of the triangle = a+b+c
Semi – perimeter=¹/₂(a+b+c)
Area of the triangle =
Prisms
A prism is polyhedron formed by two equal
parallel regular polygons, end faces connected
by side faces which are either rectangles or
parallelograms.
Types of prism
⊗ Lateral surface area of prism(L.S.A)= perimeter of cross-section * height of prism
⊗Total surface area of the prism=L.S.A+ 2A
⊗Volume of prism
= Area of cross-section * height of prism
Cylinder
Let r be the radius and h be the height of the cylinder.
⊕Lateral(curved) surface area of the cylinder
=perimeter of the base * height of cylinder.
= 2πrh
⊕Total surface area of the cylinder=L.S.A+ 2A
= 2πrh + 2π r
=2πr(r+h)
⊕Volume of cylinder =Area of the base * height
= πr²h
Sphere
A sphere is a perfectly round geometrical object
in three-dimensional space that is the surface
of a completely round ball.
⊗Surface area of the sphere = 4πr²
⊗Volume of the sphere = ⁴/₃πr³
Hemisphere
⊗Lateral〈curved〉 surface area of the hemisphere
= 2πr²
⊗Total surface area of the hemisphere = 3πr²
⊗Volume of the hemisphere = ²/₃πr³
Pyramid
A pyramid is a three-dimensional shape whose
base is a polygon. Each corner of a polygon is
attached to a singular apex, which gives the
pyramid its distinctive shape. Each base edge
and the apex form a triangle.
Types of pyramid
⊗Total surface area of the pyramid = Base area +
sum of the areas of the all the triangular faces.
i.e. A + P*l
Where P = perimeter of the base and
l is slant height.
⊗Volume of pyramid
= ¹/₃base area * height of the pyramid
Cone
A cone is a three-dimensional geometric shape
that tapers smoothly from a flat base to a point
called the apex or vertex.
⊗Curved surface area of the cone =πrl
⊗Total surface area of cone =π r (r +l)
⊗Volume of the cone = ¹/₃πr²h
Examples
1. Find the volume of the triangular prism.
soln
Area of triangular base = ¹/₂base *height
= ¹/₂ 3cm*4cm = 6cm²
Volume of prism =Area of triangular base * height of prism
= 6cm² * 10cm= 60 cm³
2. Find the volume of given hemisphere.
Soln
r = 21 cm
Volume of hemisphere (V)= ²/₃πr³
=²/₃ײ²/₇×(21cm)³
=19404 cm³
3. Given solid is made up of cone and the cylinder. The base area of the cylinder is 100 sq. cm and height of the cylinder is 3 cm . If the
volume of the whole solid is 600cm³ . Find
the height of the solid.
Soln
Base area of the cylinder = 100cm²
πr² = 100cm²
r = 5.64 cm
Heightof the cylinder ⁽h⁾= 3 cm
Volume of the whole solid = 600 cm³
Volume of the cylinder + volume of the cone
= 600
πr² h + ¹/₃π r²h’ = 600 [where h’ is the height of
the cone]
100 *3 +¹/₃*100 *h’ =600
h’= 9 cm
Total height = h+ h’ = 9cm+3cm=12cm
4. In the adjoining figure , the solid pyramid
having a square base has length of its base30
cm and height 20cm . Find the volume and total
surface area of it .
Soln
Volume of the pyramid=Area of the base * height
= ¹/₃*30cm*30cm*20cm = 6000cm³
Slant height (l) = = 25cm
Surface area of the pyramid
=area of base+¹/₂perimeter of base*slant height
= 30cm*30cm + ¹/₂*4*30cm *25cm
=2400cm²