CBSE Class 10 Mathematics Surface Areas and Volumes Notes Set 11

Cuboid and Cube

For a cuboid of length l, breadth b, height h: volume \( lbh \), total surface area \( 2(lb+bh+hl) \), lateral surface area \( 2h(l+b) \), diagonal \( \sqrt{l^2+b^2+h^2} \). A cube of edge a has volume \( a^3 \), total surface area \( 6a^2 \) and diagonal \( a\sqrt3 \).

Cylinder

Radius r and height h: curved surface area \( 2\pi rh \), total surface area \( 2\pi r(r+h) \), volume \( \pi r^2h \). A hollow cylinder of outer radius R and inner radius r has volume \( \pi h(R^2-r^2) \).

Cone and Frustum

Slant height \( l = \sqrt{r^2+h^2} \). Curved surface area \( \pi rl \), total surface area \( \pi r(r+l) \), volume \( \frac13\pi r^2h \). A frustum with radii R and r and height h has volume \( \frac13\pi h(R^2+r^2+Rr) \) and curved surface area \( \pi(R+r)l \) with \( l = \sqrt{h^2+(R-r)^2} \). A plane parallel to the base cuts a similar small cone; volumes of similar cones are in the cube of the ratio of their heights.

Sphere and Hemisphere

Sphere: surface area \( 4\pi r^2 \) and volume \( \frac43\pi r^3 \). Hemisphere: curved surface area \( 2\pi r^2 \), total surface area \( 3\pi r^2 \), volume \( \frac23\pi r^3 \).

Conversion of Solids

When one solid is melted and recast into another, the volume stays the same. Number of small solids = volume of the large solid ÷ volume of one small solid. 1 m³ = 1000 litres.

Solved Examples

Example 1. A cuboid is 12 cm × 9 cm × 8 cm. Find its total surface area and diagonal.
Answer:
TSA \( = 2(108+72+96) = 552 \) cm². Diagonal \( = \sqrt{144+81+64} = 17 \) cm.
552 cm² and 17 cm

Example 2. A cylinder has radius 7 cm and height 10 cm. Find its volume (use \( \pi = \frac{22}{7} \)).
Answer:
\( V = \frac{22}{7}\times49\times10 = 1540 \).
1540 cm³

Example 3. A cone has radius 6 cm and height 8 cm. Find its curved surface area and volume (use \( \pi = 3.14 \)).
Answer:
\( l = 10 \) cm. CSA \( = 3.14\times6\times10 = 188.4 \). Volume \( = \frac13\times3.14\times36\times8 = 301.44 \).
188.4 cm² and 301.44 cm³

Example 4. A metal sphere of radius 3 cm is melted and recast into a cylinder of radius 3 cm. Find the height of the cylinder.
Answer:
\( \frac43\pi(27) = \pi(9)h \Rightarrow h = 4 \).
4 cm

Example 5. Find the total surface area of a hemisphere of radius 7 cm (use \( \pi = \frac{22}{7} \)).
Answer:
\( 3\pi r^2 = 3\times\frac{22}{7}\times49 = 462 \).
462 cm²

Example 6. A frustum has radii 10 cm and 4 cm and height 8 cm. Find its slant height and volume (use \( \pi = 3.14 \)).
Answer:
\( l = \sqrt{64+36} = 10 \) cm. Volume \( = \frac13\times3.14\times8\times(100+16+40) = 1306.24 \).
10 cm and 1306.24 cm³

Example 7. How many cubes of edge 2 cm can be cut from a cuboid 10 cm × 8 cm × 6 cm?
Answer:
\( \frac{480}{8} = 60 \).
60 cubes

Example 8. A rectangular tank 6 m × 5 m × 4 m is filled with water. How many litres of water does it hold?
Answer:
Volume \( = 120 \) m³ \( = 120000 \) litres.
120000 litres