# 10th Class Mathematics Surface Areas and Volumes

Surface Areas and Volumes

Category : 10th Class

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

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