क्षेत्रमिति - Mensuration

क्षेत्रमिति - Mensuration

81. (i) Area of triangle = 1/2 × base × altitude

(ii) Area of triangle using heron’s formula= 

82. In an equilateral triangle with side a, then

                        where A ® Area of triangle

                        P ® Perimeter

                        h ® Height

 

83. In an isosceles triangle PQR

88. In a circle with radius r

 A/C=D/4 Where A - Area of cricle

C - Circumference of circle

D - Diameter of circle

90. (i) A circle with largest area inscribed in a right angle triangle, then r =

(ii) If ABC is an equilateral triangle with side a, then Area of circle =

(iii) If ABC is an equilateral triangle with side a, then area of circle =

(iv) If DABC is an equilateral triangle, and two circles with radius r and R, then= 

(v) Three equal circle with radius r and an equilateral triangle ABC, then area of shaded region =

91. ABCD is a square placed inside a circle with side a and radius of circle r, then 

92. Diagonal of a cube= √3✖side

 

93. Diagonal of a cuboid 

94. For two cubes 

where A1, A2 ⇒ Area of cubes

v1, v2 ⇒ Volume

a1, a2 ⇒ Sides

d1, d2 ⇒ Diagonals

95.  Units of Measurement of Area and Volume

The inter-relationships between various units of measurement of length, area and volume are listed below for ready reference:

Length

1 Centimetre (cm)   =10 milimetre (mm)

1 Decimetre (dm)  =10 centimetre

1 Metre (m)  =10 dm = 100 cm = 1000 mm

1 Decametre (dam)  =10 m = 1000 cm

1 Hectometre (hm)  =10 dam = 100 m

1 Kilometre (km)  =1000 m = 100 dam = 10 hm

1 Myriametre  =10 kilometre

Area

1 cm2= 1 cm × 1 cm   =10 mm × 10 mm = 100 mm2

1 dm2 = 1 dm × 1 dm   =10 cm × 10 cm = 100 cm2

1 m2 = 1 m × 1 m   =10 dm × 10 dm = 100 dm2

1 dam2 or 1 are =1 dam × 1dam   = 10 m × 10 m = 100 m2

1 hm2 = 1 hectare =1 hm × 1 hm   = 100 m × 10000m2 = 100 dm2

1 km2 = 1 km × 1 km   =10 hm × 10 hm = 100 hm2 or 100 hectare

Volume

1 cm3 = 1 ml  =1 cm × 1 cm × 1 cm = 10 mm × 10 mm × 10 mm = 1000 mm3

1 litre = 1000 ml  =1000 cm3

1 m3 = 1 m×1 m×1m =100 cm × 100 cm × 100 cm = 106 cm3

 =1000 litre = 1 kilometre

1 dm3 = 1000 cm3, 1m3 =1000 dm3, 1 km3 = 109 m3

If a, b, c are the edges of a cuboid, then

96. The longest diagonal= 

(i) If the height of a cuboid is zero it becomes a rectangle.

(ii) If “a” be the edge of a cube, then

(iii) The longest diagonal = a√3

97. Volume of pyramid = 1/3 ✖Base✖Area✖height (H)

98. (i) If A1& A2 denote the areas of two similar figures and l1 & l2 denote their corresponding linear measures, then= 

(ii) If V1 & V2 denote the volumes of two similar solids and l1 , l2 denote their corresponding linear measures, then= 

(iii) The rise or fall of liquid level in a container = Total volume of objects submerged or taken out Cross sectional area of container= 

99. If a largest possible cube is inscribed in a sphere of radius ‘a’ cm, then 

(i) the edge of the cube = 

(ii)  If a largest possible sphere is inscribed in a cylinder of radius ‘a’ cm and height ‘h’ cm, then for h > a,

a. the radius of the sphere = a and

b. the radius = h/2  (for a > h)

(iii)  If a largest possible sphere is inscribed in a cone of radius ‘a’ cm and slant height equal to the diameter of the base, then

a. the radius of the sphere= a/√3. 

(iv) If a largest possible cone is inscribed in a cylinder of radius ‘a’ cm and height ‘h’ cm, then the radius of the cone = a and height = h.

(v)  If a largest possible cube is inscribed in a hemisphere of radius ‘a’ cm, then the edge of the cube =a√2/3

100. In any quadrilateral

Area = 1/2✖one diagonal✖(sum of perpendiculars to it from opposite vertices) = 1/2  × d (d1 + d2)

(ii) Area of a cyclic quadrilateral= where a, b, c, d are sides of quadrilateral and

s = semi perimeter=  

101. If length, breadth & height of a three dimensional figure increase/decrease by x%, y% and z%, then

Change in area= 

Change in Volume =