Question

The height of a cone is 20 cm and its base radius is 6 cm. Find the curved surface area and the total surface area of the cone.

Answer 1

Given :-

• The height of a cone is 20 cm and its base radius is 6 cm.

To find :-

• Curved surface area and the total surface area of the cone.

Solution :-

→ Slant height = √(height)² + (radius)²

→ slant height = √h² + r²

→ l = √(20)² + (6)²

→ l = √400 + 36

→ l = √436

→ l = 20.8cm

Slant height of cone is 20.8cm

→ Curved surface area of cone

→ πrl

Put the all values

→ 22/7 × 6 × 20.8

→ 2745.6/7

→ 392.22cm² --(i)

Hence, curved surface area of cone is 10.89cm²

→ Total surface area of cone

→ πr² + πrl

→ 22/7 × (6)² + 392.22 (using i)

→ 22/7 × 36 + 392.22

→ 792/7 + 392.22

→ 113.14 + 392.22

→ 505.36 cm²

Therefore,

• Curved surface area of cone = 392.22cm²

• Total surface area of cone = 505.36cm²

Answer 2

$\tt {\pink{Given}}\begin{cases} \sf{\green{Height \: of \: the \: cone=20 \: cm}}\\ \sf{\blue{Radius \: of \: base=6 \: cm}}\\ \sf{\orange{Curved \: Surface \: Area \: of \: cone= \: ?}}\\ \sf{\red{Total \: Surface \: Area \: of \: cone= \: ?}}\end{cases}$

❍ Diagram :

$\setlength{\unitlength}{30} \begin{picture}(20,10) \linethickness{2.5} \qbezier(1,1)(3., 0)(5,1)\qbezier(1,1)(3.,2)(5,1)\put(3,1){\circle*{0.15}}\put(3,1){\line(0,1){3}}\qbezier(1,1)(1,1)(3,4)\qbezier(5,1)(3,4)(3,4)\put(3,1){\line(1,0){2}}\put(3,0.1){ \sf Radius = 6 \: cm }\put(-1,2.9){ \sf Height=20\:cm }\end{picture}$

______________________

Answer:

$\dashrightarrow\:\: \sf Slant \: Height = \sqrt{(Height)^2 + (Radius)^2}$

$\dashrightarrow\:\: \sf Slant \: Height = \sqrt{(20)^2 + (6)^2}$

$\dashrightarrow\:\: \sf Slant \: Height = \sqrt{400 + 36}$

$\dashrightarrow\:\: \sf Slant \: Height = \sqrt{436}$

$\dashrightarrow\:\: \sf Slant \: Height = 20.8 \: cm$

$\underline{\textsf{Hence, slant height of the cone is 20.8 cm}}.$

_____________________

$:\implies\sf Curved \: Surface \: Area \: of \: cone = \pi rl$

$:\implies\sf Curved \: Surface \: Area \: of \: cone = 3.14 \times 6 \times 20.8$

$:\implies\underline{\boxed{\sf Curved \: Surface \: Area \: of \: cone = 391.872 \: {cm}^{2}}}$

$\underline{\textsf{Hence, Curved Surface Area of the cone is \textbf{391.872 cm ^2 }}}.$

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$\leadsto\sf Total \: Surface \: Area \: of \: cone = \pi r \: (r+l)$

$\leadsto\sf Total \: Surface \: Area \: of \: cone = 3.14 \times 6 \: (6+20.8)$

$\leadsto\sf Total \: Surface \: Area \: of \: cone = 18.84 \times 26.8$

$\leadsto\underline{\boxed{\sf Total \: Surface \: Area \: of \: cone = 509.912 \: {cm}^{2}}}$

$\underline{\textsf{Hence, Total Surface Area of the cone is \textbf{509.912 cm ^2 }}}.$

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$\boxed{\bigstar{\sf \ Cylinder :- }}\\ \\\sf {\textcircled{\footnotesize1}} Volume \ of \ Cylinder= \pi r^2 h \\ \\ \\ \sf {\textcircled{\footnotesize2}}\ Curved \ surface\ Area \ of \ cylinder= 2\pi r h\\ \\ \\ \sf {\textcircled{\footnotesize3}} Total \ surface \ Area \ of \ cylinder= 2\pi r (h+r)$

$\boxed{\bigstar{\sf \ Cone :- }}\\ \\\sf {\textcircled{\footnotesize1}} Volume \ of \ Cone= \dfrac{1}{3}\pi r^2 h \\ \\ \\ \sf {\textcircled{\footnotesize2}}\ Curved \ surface\ Area \ of \ Cone = \pi r l \\ \\ \\ \sf {\textcircled{\footnotesize3}} Total \ surface \ Area \ of \ Cone = \pi r (l+r) \\ \\ \\ \sf {\textcircled{\footnotesize4}} Slant \ Height \ of \ cone (l)= \sqrt{r^2+h^2}$

$\boxed{\bigstar{\sf \ Hemisphere :- }}\\ \\\sf {\textcircled{\footnotesize1}} Volume \ of \ Hemisphere= \dfrac{2}{3}\pi r^3 \\ \\ \\ \sf {\textcircled{\footnotesize2}}\ Curved \ surface\ Area \ of \ Hemisphere = 2 \pi r^2 \\ \\ \\ \sf {\textcircled{\footnotesize3}} Total \ surface \ Area \ of \ Hemisphere = 3 \pi r^2$

$\boxed{\bigstar{\sf \ Sphere :- }}\\ \\\sf {\textcircled{\footnotesize1}} Volume \ of \ Sphere= \dfrac{4}{3}\pi r^3 \\ \\ \\ \sf {\textcircled{\footnotesize2}}\ Surface\ Area \ of \ Sphere = 4 \pi r^2$