Question

Find the volume of the sphere in terms of $\pi$

Answer 1

check the picture below.

so we use the blue triangle to get the slanted length of the isosceles triangle, that means the upper-tangents are each 13 - 5, as you see there, then we use the yellow triangle with those lengths, and the pythagorean theorem to get r.

$\bf \stackrel{\textit{volume of a sphere}}{V=\cfrac{4\pi r^3}{3}}\implies V=\cfrac{4\pi \left( \frac{10}{3} \right)^3}{3}\implies V=\cfrac{4\pi \cdot \frac{10^3}{3^3}}{3} \\\\\\ V=\cfrac{\frac{4000\pi }{27}}{3}\implies V=\cfrac{4000\pi }{81}\implies V=49\frac{31}{81}\pi$

Answer 2

Look at the picture.

Use the Pythagorean theorem to calculate the length of a side x:

$x^2=5^2+12^2\\\\x^2=25+144\\\\x^2=169\to x=\sqrt{169}\to x=13\ cm$

Use the formula of a radius of a circle inscribed in a triangle:

$r=\dfrac{2A_\Delta}{a+b+c}$

$A_\Delta$ - an area of a triangle

$a,\ b,\ c$ - the sides of the triangle

Calculate:

$A_\Delta=\dfrac{10\cdot12}{2}=60\ cm^2$

$r=\dfrac{2\cdot60}{13+13+10}=\dfrac{120}{36}=\dfrac{10}{3}\ cm$

The fromula of the volume of a sphere:

$V=\dfrac{4}{3}\pi r^3$

Substitute:

$V=\dfrac{4}{3}\pi\cdot\left(\dfrac{10}{3}\right)^3=\dfrac{4}{3}\pi\cdot\dfrac{1,000}{27}=\dfrac{4,000\pi}{81}\ cm^3\approx49\pi\ cm^3$