Mensuration

Area of Triangle

Area of the Area = 12$\frac{1}{2}$base*height

 If a , b c be the                 the sides of the traingle , then perimeter of the triangle = a + b+ c

$$Semi – perimeter=a+b+c2$\frac{\mathrm{a}+\mathrm{b}+\mathrm{c}}{2}$

Area of the triangle = s(s−a)(s−b)(s−c)−−−−−−−−−−−−−−−−−√$\sqrt{\mathrm{s}\left(\mathrm{s}-\mathrm{a}\right)\left(\mathrm{s}-\mathrm{b}\right)\left(\mathrm{s}-\mathrm{c}\right)}$

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 the base* height of prism

Total surfaca area of the prism=L.S.A+ 2A

Volume of prism = l*b*h= Area of the base * height

 Cylinders

Let `r’ be the radius of the cylinder and`h’ be the height of the cylinder.

Lateral surface area of the cylinder =perimeter of the base* height of prism

= 2πrh

Total surface area of the cylinder=L.S.A+ 2A

= 2πrh + 2π r2

 =2πr(r+h)

Volume of the 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 =43$\frac{4}{3}$πr3

Hemisphere

Lateral surface area of the hemisphere = 2πr2

Total surface area of the hemisphere = 3πr2

Volume of the hemisphere = 23$\frac{2}{3}$πr2

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.

= A + 12$\frac{1}{2}$ P*L

Where P = perimeter of the base and l is slanting height.

Volume of pyramid = 13$\frac{1}{3}$base 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 =13$\frac{1}{3}$ π r2 h

Examples

1. Find the volume of the triangular prism.

soln

Area of the base of the triangle = 12base *height$$

= 123*4$$= 6cm2

Volume of prism = 12base *height * height of prism

= 6* 10= 60 cm2

2.

Soln

r =422$\frac{42}{2}$ = 21 cm

Volume of hemisphere (V)=23πr3$\frac{2}{3}\pi {\mathrm{r}}^3$=23∗227213$\frac{2}{3}\ast \frac{22}{7}{21}^3$ =19404 cm2

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 600 cubic cm . Find the height of the solid .

Soln

Base area of the cylinder = 100

πr2 = 100

r = 5.64 cm

Heightof the cylinder = 3 cm

Volume of the whole solid = 600 sq . cm

Volume of the cylinder + volume of the cone = 600

πr2h + 13$\frac{1}{3}$π r2h’ = 600 [where h’ is the height of the cone]

100 *3 + 13$\frac{1}{3}$ *100 *h’ =600

h’= 9 cm

 Total height = h+ h’ = 9+3=12cm

4. In the adjoining figure , the solid pyramid having a square base has length of its base30 cmand height 20cm . Find the volume and total surface area of it .

Soln

Volume of the pyramid =13$\frac{1}{3}$Area of the base * height$$

= 13$\frac{1}{3}$ 30*30 *20 = 6000cm3

Slant height (l) = 202+152−−−−−−−−√$\sqrt{{20}^2+{15}^2}$= 25cm

Surface area of the pyramid = A + 12P∗L$\frac{1}{2}\mathrm{P}\ast \mathrm{L}$

= 30*30 + 12$\frac{1}{2}$ *4*30 *25

=2400cm2