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Mathematics
Mensuration Problems on Areas and Volumes – Formulas, Trigonometry Basics, Questions and Solved Examples
Diameter, D = 2R
Area = πR2 sq. units
Circumference = 2πR unitsg
Square:
Area = a2 sq. units
Perimeter = 4a units
Diagonal, d = √2 a units
Rectangle:
Area = L*B sq. units
Perimeter = 2(L+B) units
Diagonal, d = √L2+B2 units
Right Angled Triangle:
Area = (½)bxh sq. units
Perimeter = b + h + hypotenuse
Hypotenuse = √b2+h2 units
Equilateral Triangle:
Area = √4 a2 sq. units
Perimeter = 3a units, where a = side of the triangle
Scalene Triangle:h
Area: √s(s-a)(s-b)(s-c) sq. units; s = (a+b+c)/2
Perimeter = (a+b+c) units
Isosceles Triangle:
Area = b/4 √4a2-B2 sq units
Perimeter = 2a + b units, where b = base length; a = equal side length
Cube:
Volume = a3 cubic units
Lateral Surface Area (LSA) = 4a2 sq. units
Total surface area (TSA) = 6a2 sq. units
Length of diagonal = a√3 units
Cuboid:
Volume = (Cross section area * height) = L * B * H cubic units
Lateral Surface Area (LSA) = 2[(L+B)H] sq. units
Total surface area (TSA) = 2(LB+BH+HL) sq. units
Length of the diagonals = √L2+B2+H2 units
Sphere:
Volume = (4/3) πR3 cubic units
Surface Area = 4πR2 sq. units
If R and r are the external and internal radii of a spherical shell, then its Volume = (4/3) [R3-r3] cubic units
Hemisphere:
Volume = (2/3) πR3 cubic units
TSA = 3πR2 sq. units
Cylinder:
Volume = πr2h cubic units
Curved surface Area (CSA) (excludes the areas of the top and bottom circular regions) = 2πRh sq. units
TSA = Curved Surface Area + Areas of the top and bottom circular regions = 2πRh + 2πR2 = 2πR[R+h] sq. units
Cone:
Volume = (1/3) πR2h cubic units
Slant Height of cone, L = √R2+H2 units
CSA = πRL sq. units
Mensuration and Trigonometry Problems (Areas and Volumes Questions)
Part 1
1A right circular cone is placed over a cylinder of the same radius. Now the combined structure is painted on all sides. Then they are separated now the ratio of area painted on Cylinder to Cone is
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