Annulus from triangle

Calculate the content of the area bounded by a circle circumscribed and a circle inscribed by a triangle with sides a = 25mm, b = 29mm, c = 36mm

Correct answer:

S =  831.0003 mm²

Step-by-step explanation:

$$\begin{gathered} a=25\text{ mm } \\ b=29\text{ mm } \\ c=36\text{ mm } \\ {c}_1=\sqrt{a^2+b^2}=\sqrt{25^2+29^2}=\sqrt{1466}\text{ mm}\doteq 38.2884\text{ mm } \\ {c}_1\mathrm{\ne }c \\ s={\displaystyle \frac{a+b+c}{2}}={\displaystyle \frac{25+29+36}{2}}=45\text{ mm } \\ A=\sqrt{s\cdot (s-a)\cdot (s-b)\cdot (s-c)}=\sqrt{45\cdot (45-25)\cdot (45-29)\cdot (45-36)}=360{\text{mm}}^2 \\ R={\displaystyle \frac{a\cdot b\cdot c}{4\cdot A}}={\displaystyle \frac{25\cdot 29\cdot 36}{4\cdot 360}}={\displaystyle \frac{145}{8}}=18.125\text{ mm } \\ r={\displaystyle \frac{A}{s}}={\displaystyle \frac{360}{45}}=8\text{ mm } \\ {S}_1=\pi \cdot R^2=3.1416\cdot 18.125^2\doteq 1032.0623{\text{mm}}^2 \\ {S}_2=\pi \cdot r^2=3.1416\cdot 8^2\doteq 201.0619{\text{mm}}^2 \\ S=S_1-S_2=1032.0623-201.0619=831.0003{\text{mm}}^2 \end{gathered}$$

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