Five-gon

Calculate the side a, the circumference and the area of the regular 5-angle if Rop = 6cm.

Correct answer:

a =  7.0534 cm
o =  35.2671 cm
S =  85.5951 cm²

Step-by-step explanation:

$$\begin{gathered} n=5 \\ R=6\text{ cm } \\ \mathrm{\Phi }=360\mathrm{/}(2\cdot n)=360\mathrm{/}(2\cdot 5)=36^{\circ } \\ \sin \mathrm{\Phi }=a\mathrm{/}2:R \\ a=2\cdot R\cdot \sin {\mathrm{\Phi }}^{\circ }=2\cdot R\cdot \sin 36^{\circ }=2\cdot 6\cdot \sin 36^{\circ }=2\cdot 6\cdot 0.587785=7.053=7.0534\text{ cm} \end{gathered}$$

$o=n\cdot a=5\cdot 7.0534=35.2671\text{ cm}$

$$\begin{gathered} h=\sqrt{R^2-(a\mathrm{/}2{)}^2}=\sqrt{6^2-(7.0534\mathrm{/}2{)}^2}\doteq 4.8541\text{ cm } \\ {S}_1={\displaystyle \frac{a\cdot h}{2}}={\displaystyle \frac{7.0534\cdot 4.8541}{2}}\doteq 17.119{\text{cm}}^2 \\ S=n\cdot S_1=5\cdot 17.119=85.5951{\text{cm}}^2 \end{gathered}$$

Try another example

Related math problems and questions:

• 6 regular polygon
  It is given 6 side regular polygon whose side is 5 cm. Calculate its content area. Compare how many more cm² (square centimeters) has a circle in which is inscribed the 6-gon.
• Pentagon
  Calculate the length of side, circumference and area of a regular pentagon, which is inscribed in a circle with radius r = 6 cm.
• n-gon II
  What is the side length of the regular 5-gon circumscribed circle of radius 11 cm?
• n-gon
  What is the side length of the regular 5-gon inscribed in a circle of radius 12 cm?
• 30-gon
  At a regular 30-gon the radius of the inscribed circle is 15cm. Find the "a" side size, circle radius "R", circumference, and content area.
• Regular 5-gon
  Calculate the area of the regular pentagon with side 7 cm.
• Regular n-gon
  Which regular polygon have a radius of circumscribed circle r = 10 cm and the radius of inscribed circle p = 9.962 cm?
• Nine-gon
  Calculate the perimeter of a regular nonagon (9-gon) inscribed in a circle with a radius 13 cm.
• Tetrahedral pyramid
  A regular tetrahedral pyramid is given. Base edge length a = 6.5 cm, side edge s = 7.5 cm. Calculate the volume and the area of its face (side area).
• Circle described
  The radius of the circle described to the right triangle with 6 cm long leg is 5 cm. Calculate the circumference of this triangle.
• Regular hexagonal pyramid
  Calculate the height of a regular hexagonal pyramid with a base edge of 5 cm and a wall height of w = 20cm. Sketch a picture.
• Hexagonal pyramid
  Calculate the volume and the surface of a regular hexagonal pyramid with a base edge length of 3 cm and a height of 5 cm.
• Wall height
  Calculate the height of a regular hexagonal pyramid with a base edge of 5 cm and a wall height w = 20 cm.
• Triangle and its heights
  Calculate the length of the sides of the triangle ABC, if v_a=5 cm, v_b=7 cm and side b is 5 cm shorter than side a.
• Hexagon
  Calculate the regular hexagonal prism's surface whose base edge a = 12cm and side edge b = 3 dm.
• Hexagonal prism
  The base of the prism is a regular hexagon consisting of six triangles with side a = 12 cm and height va = 10.4 cm. The prism height is 5 cm. Find the volume and surface of the prism.
• 9-gon pyramid
  Calculate the volume and the surface of a nine-sided pyramid, the base of which can be inscribed with a circle with radius ρ = 7.2 cm and whose side edge s = 10.9 cm.