10th Class Mathematics Surface Areas and Volumes

Surface Area and Volume

• A right circular cylinder is a solid generated by the revolution of a rectangle about one of its sides.

(i) Volume\[(V)=\pi {{r}^{2}}h\]    

(ii) Curved surface area (C.S.A.) \[=2\pi \,rh\]         

(iii) Base area\[=\pi \,{{r}^{2}}\]       

(iv) Total surface area (T.S.A.) \[=2\pi \,r(h+r)\]

• Hollow cylinders:  

(i) Volume of material \[=\pi h({{R}^{2}}-r)\]                       

(ii) Curved surface area \[=2\pi h(R+r)\]                                             

(iii) Total surface area \[=2\pi (R+r)(h+R-r)\]

• Right circular cone: If a right angled triangle is revolved about one of the sides containing the right angle, the solid thus generated is called a right circular cone.

(i) \[l=\sqrt{{{h}^{2}}+{{r}^{2}}}\]

(ii) \[volume=\frac{1}{3}\pi {{r}^{2}}h\]

(iii) \[C.S.A.=\pi rl=\pi r\sqrt{{{h}^{2}}+{{r}^{2}}}\]

(iv) \[T.S.A.=\pi r(l+r)\]

• Sphere: A sphere is the locus of a point which moves in space in such a way that its distance from a fixed point remains constant.
• (i) Volume\[=\frac{4}{3}\pi {{r}^{3}}\]
• (ii) Surface area\[=4\pi {{r}^{2}}\]

• Hemisphere: A plane through the centre of a sphere cuts it into two equal parts. Each part is called a hemisphere.

(i) Volume\[=\frac{2}{3}\pi {{r}^{3}}\]

(ii) Curved surface area\[=2\pi {{r}^{2}}\]

(iii) Total surface area\[=3\pi {{r}^{2}}\]

Volume and surface area of a frustum of a cone: When a cone is cut by a plane parallel to its base, the portion between the plane and the base is called frustum of the cone.

(i) \[V=\frac{\pi h}{3}[{{R}^{2}}+{{r}^{2}}+Rr]\]

(ii) \[L.S.A=\pi l[R+r]\]

(iii) \[T.S.A=\pi [{{R}^{2}}+{{r}^{2}}+l(R+r)]\]

Where, \[l=\sqrt{{{h}^{2}}+{{(R-r)}^{2}}}\]