Question

if vertices of regular pentagon of perimeter 50 cm on the circumference of circle find the radius of the circle​

Answer 1

If vertices of regular pentagon of perimeter 50 cm are on the circumference of a circle then the radius of the circle is 8.21 cm

Step-by-step explanation:

As shown in the figure, ABCDE is a regular pentagon

Since the perimeter of the pentagon is 50 cm

Therefore, each side of the pentagon will be 50/5 = 10 cm

The circumcirle of the pentagon will also be the circumcircle of triangles ABC, ADC and AED

Since the internal angle of a regular pentagon is 105°

Therefore,

∠ABC = 105°

∴ ∠BAC = ∠BCA = 75°/2 = 37.5°

Also AB = BC = 10 cm

In ΔABC

By sine rule

$\frac{\sin 105^\circ}{AC}=\frac{\sin 37.5^\circ}{AB}$

$\implies AC=\frac{AB\sin 105^\circ}{\sin 37.5^\circ}$

$\implies AC=10\times\frac{0.966}{0.609}$

$\implies AC=15.86$ cm

Now the area of ΔABC

$\Delta=\frac{1}{2}\timesAB\times BC\times\sin 105^\circ$

$\implies \Delta=\frac{1}{2}\times 10\times 10\times 0.966$

$\implies \Delta=48.3$ cm²

We know that radius of a circumcircle of a triangle is given by

$\boxed{R=\frac{abc}{4\Delta}}$

$\implies R=\frac{10\times 10\times 15.86}{4\times 48.3}$

$\implies R=8.209$ cm

Thus, the radius of the circle is 8.21 cm

Hope this answer is helpful.

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