CBSE Class 10 Mathematics Surface Areas and Volumes Notes Set 10

Surface Areas of Solids

The curved (lateral) surface of a cylinder unrolls into a rectangle, so its area is \( 2\pi rh \), and with both circular ends the total is \( 2\pi r(r+h) \). A cone has curved area \( \pi rl \), where the slant height is \( l=\sqrt{r^2+h^2} \), and total area \( \pi r(l+r) \). A sphere has area \( 4\pi r^2 \); a hemisphere has curved area \( 2\pi r^2 \) and total area \( 3\pi r^2 \).
A cuboid of sides l, b, h has total surface \( 2(lb+bh+hl) \), lateral surface \( 2h(l+b) \) and diagonal \( \sqrt{l^2+b^2+h^2} \). A cube of edge a has total surface \( 6a^2 \) and diagonal \( a\sqrt3 \).

Volumes of Solids

Cuboid \( lbh \), cube \( a^3 \), cylinder \( \pi r^2h \), cone \( \frac13\pi r^2h \), sphere \( \frac43\pi r^3 \), hemisphere \( \frac23\pi r^3 \). Notice that a cone, a hemisphere and a cylinder with equal radius r and height r have volumes in the ratio 1 : 2 : 3.
Whenever a solid is melted and recast, the volume is conserved. The number of small pieces equals the volume of the large solid divided by the volume of one piece. For a liquid flowing through a pipe, the volume delivered in a given time is the cross-section area multiplied by the length of the water column (speed x time).

Combination of Solids and the Frustum

For a toy or a tent made of several solids, add the volumes of the parts, but for the surface add only the surfaces that are visible; the faces where two solids are joined disappear. When a cone is cut by a plane parallel to its base, the part below the plane is a frustum. If its end radii are R and r, height h and slant height \( l=\sqrt{h^2+(R-r)^2} \), then volume \( =\frac13\pi h(R^2+r^2+Rr) \), curved area \( =\pi(R+r)l \) and total area \( =\pi(R+r)l+\pi R^2+\pi r^2 \).
The small cone removed is similar to the whole cone, so if its height is k times the whole height, its volume is k³ times the whole volume.