# Tetrahedral pyramid

Calculate the volume and surface of the regular tetrahedral pyramid if content area of the base is 20 cm2 and deviation angle of the side edges from the plane of the base is 60 degrees.

Result

V =  36.51 cm3
S =  72.915 cm2

#### Solution:

$S_{1}=20 \ \text{cm}^2 \ \\ Φ=60 \ ^\circ \ \\ \ \\ S_{1}=a^2 \ \\ a=\sqrt{ S_{1} }=\sqrt{ 20 } \doteq 2 \ \sqrt{ 5 } \ \text{cm} \doteq 4.4721 \ \text{cm} \ \\ \ \\ u=\sqrt{ 2 } \cdot \ a=\sqrt{ 2 } \cdot \ 4.4721 \doteq 2 \ \sqrt{ 10 } \ \text{cm} \doteq 6.3246 \ \text{cm} \ \\ \ \\ h=\dfrac{ u }{ 2 } \cdot \ \tan Φ ^\circ =\dfrac{ u }{ 2 } \cdot \ \tan 60^\circ \ =\dfrac{ 6.32455532034 }{ 2 } \cdot \ \tan 60^\circ \ =\dfrac{ u }{ 2 } \cdot \ 1.732051=5.47723 \ \\ \ \\ V=\dfrac{ 1 }{ 3 } \cdot \ S_{1} \cdot \ h=\dfrac{ 1 }{ 3 } \cdot \ 20 \cdot \ 5.4772 \doteq 36.5148 \doteq 36.51 \ \text{cm}^3$
$b=\sqrt{ h^2 + (a/2)^2 }=\sqrt{ 5.4772^2 + (4.4721/2)^2 } \doteq \sqrt{ 35 } \ \text{cm} \doteq 5.9161 \ \text{cm} \ \\ \ \\ S_{2}=a \cdot \ b/2=4.4721 \cdot \ 5.9161/2 \doteq 5 \ \sqrt{ 7 } \ \text{cm}^2 \doteq 13.2288 \ \text{cm}^2 \ \\ \ \\ S=4 \cdot \ S_{2} + S_{1}=4 \cdot \ 13.2288 + 20 \doteq 72.915 \doteq 72.915 \ \text{cm}^2$

Our examples were largely sent or created by pupils and students themselves. Therefore, we would be pleased if you could send us any errors you found, spelling mistakes, or rephasing the example. Thank you!

Leave us a comment of this math problem and its solution (i.e. if it is still somewhat unclear...):

Be the first to comment!

Tips to related online calculators
Pythagorean theorem is the base for the right triangle calculator.
Tip: Our volume units converter will help you with the conversion of volume units.

## Next similar math problems:

1. Tetrahedral pyramid
Calculate the surface S and the volume V of a regular tetrahedral pyramid with the base side a = 5 m and a body height of 14 m.
2. 4side pyramid
Calculate the volume and surface of 4 sides regular pyramid whose base edge is 4 cm long. The angle from the plane of the sidewall and base plane is 60 degrees.
Given is a regular quadrangular pyramid with a square base. The body height is 30 cm and volume V = 1000 cm³. Calculate its side a and its surface area.
4. 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.
5. The plaster cast
The plaster cast has the shape of a regular quadrilateral pyramid. The cover consists of four equilateral triangles with a 5 m side. Calculate its volume and surface area.
6. Pyramid 8
Calculate the volume and the surface area of a regular quadrangular pyramid with the base side 9 cm and side wall with the base has an angle 75°.
7. Right pyramid
A right pyramid on a base 4 cm square has a slant edge of 6 cm. Calculate the volume of the pyramid.
How many square meters is needed to cover the tower the shape of regular quadrangular pyramid base edge 10 meters, if the deviation lateral edges from the base plane is 68 °? Calculate coverage waste 10%.
9. Pyramid - angle
Calculate the surface of regular quadrangular pyramid whose base edge measured 6 cm and the deviation from the plane of the side wall plane of the base is 50 degrees.
10. 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)?