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:

S1=20 cm2 Φ=60 ∘  S1=a2 a=S1=20≐2 5 cm≐4.4721 cm  u=2⋅ a=2⋅ 4.4721≐2 10 cm≐6.3246 cm  h=u2⋅ tan⁡Φ∘=u2⋅ tan⁡60∘ =6.324555320342⋅ tan⁡60∘ =u2⋅ 1.732051=5.47723  V=13⋅ S1⋅ h=13⋅ 20⋅ 5.4772≐36.5148≐36.51 cm3S_{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=h2+(a/2)2=5.47722+(4.4721/2)2≐35 cm≐5.9161 cm  S2=a⋅ b/2=4.4721⋅ 5.9161/2≐5 7 cm2≐13.2288 cm2  S=4⋅ S2+S1=4⋅ 13.2288+20≐72.915≐72.915 cm2b=\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

Try another example


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...):

Showing 0 comments:
1st comment
Be the first to comment!
avatar




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.
See also our trigonometric triangle calculator.

You need to know the following knowledge to solve this word math problem:


 

 

Next similar math problems:

  1. Tetrahedral pyramid
    jehlan_4b_obdelnik_3 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
    ihlany 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.
  3. Quadrangular pyramid
    jehlan_4b_obdelnik_1 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
    pyramid_2 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
    pyramid_4s 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
    ihlan 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
    squarepyramid A right pyramid on a base 4 cm square has a slant edge of 6 cm. Calculate the volume of the pyramid.
  8. Regular quadrangular pyramid
    6 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
    pyramid 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
    3sidesPyramid 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)?
  11. Regular quadrilateral pyramid
    pyramid_3 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.
  12. Flowerbed
    5928-vyvyseny-zahon-2 Flowerbed has the shape of a truncated pyramid, the bottom edge of the base a = 10 m, the upper base b = 9 m. Deviation angle between edge and the base is alpha = 45°. What volume is needed to make this flowerbed? How many plants can be planted if 1 m2 =
  13. Tetrahedron
    tetrahedron (1) Calculate height and volume of a regular tetrahedron whose edge has a length 18 cm.
  14. Reference angle
    anglemeter Find the reference angle of each angle:
  15. Maple
    tree_javor Maple peak is visible from a distance 3 m from the trunk from a height of 1.8 m at angle 62°. Determine the height of the maple.
  16. If the
    tan If the tangent of an angle of a right angled triangle is 0.8. Then its longest side is. .. .
  17. Theorem prove
    thales_1 We want to prove the sentence: If the natural number n is divisible by six, then n is divisible by three. From what assumption we started?