Answer to Question #206769 in Geometry for Shane

Question

The volume of a right circular cone obtained from a sector of a circle in which the radius is 26 cm and and the central angle is 138.5

Answer 1

Length of the arc is

$l=\frac{\theta}{360 \degree }\times 2\pi r\\ \theta=138.5\degree\\ r=26cm\\ l=\frac{138.5\degree}{360 \degree }\times 2\times \frac{22}{7}\times 26cm\\ l=62.9cm\\$

Length of arc is the same as the circumference of base

$l=2\pi r_b\\ r_b=\frac{l}{2\pi}\\ r_b=\frac{62.9}{2\pi}\\ r_b=10cm\\$

Slant height, cone height and base radius form a right-angle triangle

$l_s=\sqrt{r_b^2+h^2}\\ l_s=r=26cm\\ h=\sqrt{26^2-10^2}\\ h=24cm\\$

Volume of cone is

$V=\frac{1}{3}\pi r_b^2h\\ V=\frac{1}{3}\times\frac{22}{7}\times{10^2}\times{24}\\ V=2514cm^3$