# Algebraic Identities

• $(a+b)(a-b)=a^{2}-b^{2}$ • $(a+b)^{2}=a^{2}+2ab+b^{2}$ • $(a-b)^{2}=a^{2}-2ab+b^{2}$ • $(x+a)(x+b)=x^{2}+x(a+b)+ab$ • $(x+a)(x+b)(x+c)=x^{3}+x^{2}(a+b+c)+x(ab+bc+ca)+abc$ • $(a+b)^{3}=a^{3}+3ab(a+b)+b^{3}$ • $(a-b)^{3}=a^{3}-3ab(a-b)-b^{3}$ • $a^{3}+b^{3}=(a+b)(a^{2}-ab+b^{2})$ • $a^{3}-b^{3}=(a-b)(a^{2}+ab+b^{2})$ • $(a+b+c)^{2}=a^{2}+b^{2}+c^{2}+2ab+2bc+2ca$ • $a^{4}+a^{2}b^{2}+b^{4}=(a^{2}+b^{2}+ab)(a^{2}+b^{2}-ab)$ • $(a+b+c)^{3}-a^{3}-b^{3}-c^{3}=3(a+b)(b+c)(c+a)$ • $(a+b)(b+c)(c+a)=(a+b+c)(ab+bc+ca)-abc$ # Mensuration

## Area and Perimeter of Plane Figures

 Name Perimeter Area Triangle (a+b+c), where a, b, c are sides ${\frac {1}{2}}bh$ where "h" is the height from any vertex to the opposite side "b" Circle $2{\pi }r$ ${\pi }r^{2}$ Square $4a$ Where a is the side of a square $a^{2}$ Rectangle $2(l+b)$ Where l & b are the length & breadth $lb$ Trapezium $(a+b+c+d)$ Where a,b,c and d are the sides ${\frac {1}{2}}h(a+b)$ Where a and b are parallel sides of trapezium. And h is the perpendicular distance between two parallel sides. Parallelogram $2(a+b)$ Where a & b are the sides of Parallelogram $bh$ Where b is base and h is the perpendicular distance between base b and its parallel side. Rhombus $4a$ Where a is the side of a rhombus ${\frac {1}{2}}(d_{1}d_{2})$ Where d1 and d2 are diagonals of rhombus

## LSA(CSA) TSA & VOLUME of Solid Figures

 Name of the Solid LSA(CSA)in sq.units TSA in sq.units VOLUME in cubic units CUBE $4l^{2}$ Where lenght(l)=breadth(b)=height(h) $6l^{2}$ $l^{3}$ CUBOID $2h(l+b)$ $2(lb+bh+lh)$ $lbh$ PRISM 1)EQUILATERAL TRIANGLE RIGHT PRISM $Ph$ Where P=3a is the perimeter of base triangle $2B+Ph$ Where B=${\frac {{\sqrt {3}}a^{2}}{4}}$ is the area of base $Bh$ Where B=${\frac {{\sqrt {3}}a^{2}}{4}}$ 2)SQUARE BASED RIGHT PRISM $Ph$ Where P=4a is the perimeter of base square $2B+Ph$ Where B=$a^{2}$ is the area of base $Bh$ Where B=$a^{2}$ PYRAMID 1)EQUILATERAL TRIANGLE BASED RIGHT PYRAMID ${\frac {1}{2}}Pl$ Where P=3a is the perimeter of base triangle l is the slant height $B+{\frac {1}{2}}Pl$ Where B=${\frac {{\sqrt {3}}a^{2}}{4}}$ is the area of base ${\frac {1}{3}}Bh$ Where B=${\frac {{\sqrt {3}}a^{2}}{4}}$ 2)SQUARE BASED RIGHT PYRAMID ${\frac {1}{2}}Pl$ Where P=4a is the perimeter of base square l is the slant height $B+{\frac {1}{2}}Pl$ Where B=$a^{2}$ is the area of base ${\frac {1}{3}}Bh$ Where B=$a^{2}$ CYLINDER $2{\pi }rh$ Where r is the radius of circular base $2{\pi }r(r+h)$ where "h" is the height of cylinder ${\pi }r^{2}h$ CONE ${\pi }rl$ Where l is the slant height ${\pi }r(l+r)$ Where r is the radius of circular base ${\frac {1}{3}}{\pi }r^{2}h$ Where h is the height or depth of the cone FRUSTUM OF CONE ${\pi }(r_{1}+r_{2})l$ Where l=${\sqrt {h^{2}+(r_{1}-r_{2})^{2}}}$ $\displaystyle π{{(r_{1}+r_{2})l+r_{1}^2+r_{2}^2}}$ Where $r_{1}$ & $r_{2}$ are the radii of two bases$(r_{1}>r_{2})$ ${\frac {1}{3}}{\pi }h(r_{1}^{2}+r_{2}^{2}+r_{1}r_{2})$ Where h is the height or depth of the frustum ofcone SPHERE $4{\pi }r^{2}$ $4{\pi }r^{2}$ ${\frac {4}{3}}{\pi }r^{3}$ HEMISPHERE $2{\pi }r^{2}$ $3{\pi }r^{2}$ ${\frac {2}{3}}{\pi }r^{3}$ 