Pentagonal pyramid

The height of a regular pentagonal pyramid is as long as the edge of the base, 20 cm. Calculate the volume and surface area of the pyramid.

Correct answer:

V =  4587.9397 cm³
S =  1902.113 cm²

Step-by-step explanation:

$$\begin{gathered} a=20\text{ cm } \\ h=a=20\text{ cm } \\ n=5 \\ r={\displaystyle \frac{\sqrt{25+10\cdot \sqrt{5}}}{10}}\cdot a={\displaystyle \frac{\sqrt{25+10\cdot \sqrt{5}}}{10}}\cdot 20\doteq 13.7638\text{ cm } \\ {S}_1={\displaystyle \frac{\sqrt{25+10\cdot \sqrt{5}}}{4}}\cdot a^2={\displaystyle \frac{\sqrt{25+10\cdot \sqrt{5}}}{4}}\cdot 20^2\doteq 688.191{\text{cm}}^2 \\ V={\displaystyle \frac{1}{3}}\cdot S_1\cdot h={\displaystyle \frac{1}{3}}\cdot 688.191\cdot 20=4587.9397{\text{cm}}^3 \end{gathered}$$

$$\begin{gathered} h_2=\sqrt{r^2+h^2}=\sqrt{13.7638^2+20^2}\doteq 24.2784\text{ cm } \\ {S}_2={\displaystyle \frac{a\cdot h_2}{2}}={\displaystyle \frac{20\cdot 24.2784}{2}}\doteq 242.7844{\text{cm}}^2 \\ S=S_1+n\cdot S_2=688.191+5\cdot 242.7844=1902.113{\text{cm}}^2 \end{gathered}$$

Try another example

Showing 1 comment:

Sebf

Thanks for the demonstration.
All my respect.
Regards.
S.

1 month ago  Like

Related math problems and questions:

• Pentagonal pyramid
  Find the volume and surface of a regular pentagonal pyramid with a base edge a = 12.8 cm and a height v = 32.1 cm.
• Hexagonal pyramid
  Find the volume of a regular hexagonal pyramid, the base edge of which is 12 cm long and the side edge 20 cm.
• Hexagonal pyramid
  Calculate the surface area of a regular hexagonal pyramid with a base inscribed in a circle with a radius of 8 cm and a height of 20 cm.
• Hexagonal pyramid
  Please calculate the height of a regular hexagonal pyramid with a base edge of 5cm and a wall height of w = 20cm. Please sketch a picture.
• Wall height
  Calculate the height of a regular hexagonal pyramid with a base edge of 5 cm and a wall height w = 20 cm.
• 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.
• Pyramid height
  Find the volume of a regular triangular pyramid with edge length a = 12cm and pyramid height h = 20cm.
• Regular quadrilateral pyramid
  What is the volume of a regular quadrilateral pyramid if its surface is 576 cm² and the base edge is 16 cm?
• Triangular pyramid
  Calculate the volume of a regular triangular pyramid with edge length a = 12cm and pyramid height v = 20cm.
• 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.
• 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.
• Truncated pyramid
  Find the volume and surface area of a regular quadrilateral truncated pyramid if base lengths a1 = 17 cm, a2 = 5 cm, height v = 8 cm.
• Quadrilateral pyramid,
  A quadrilateral pyramid, which has a rectangular base with dimensions of 24 cm, 13 cm. The height of the pyramid is 18cm. Calculate 1/the area of the base 2/casing area 3/pyramid surface 4/volume of the pyramid
• Quadrangular pyramid
  The regular quadrangular pyramid has a base length of 6 cm and a side edge length of 9 centimeters. Calculate its volume and surface area.
• Pyramid 4sides
  Calculate the volume and the surface of a regular quadrangular pyramid when the edge of the base is 4 cm long and the height of the pyramid is 7 cm.
• Triangular pyramid
  It is given perpendicular regular triangular pyramid: base side a = 5 cm, height v = 8 cm, volume V = 28.8 cm³. What is it content (surface area)?
• Regular quadrilateral pyramid
  Find the volume and surface of a regular quadrilateral pyramid if the bottom edge is 45 cm long and the pyramid height is 7 cm.