Regular hexagonal prism

Calculate the volume of a regular hexagonal prism whose body diagonals are 24cm and 25cm long.

Correct answer:

V =  2636.7965 cm³

Step-by-step explanation:

$$\begin{gathered} u=24\text{ cm } \\ v=25\text{ cm } \\ {u}_1=2a \\ (u_2\mathrm{/}2{)}^2+(a\mathrm{/}2{)}^2=a^2 \\ {u}_2^2\mathrm{/}4+a^2\mathrm{/}4=a^2 \\ {u}_2^2\mathrm{/}4+=3\mathrm{/}4a^2 \\ {u}_2=\sqrt{3}\cdot a \\ {h}^2+u_1^2=v^2 \\ {h}^2+u_2^2=u^2 \\ {h}^2+(2a{)}^2=v^2 \\ {h}^2+(\sqrt{3}\cdot a{)}^2=u^2 \\ 4a^2-3a^2=v^2-u^2 \\ {a}^2=v^2-u^2 \\ a=\sqrt{v^2-u^2}=\sqrt{25^2-24^2}=7\text{ cm } \\ h=\sqrt{v^2-4\cdot a^2}=\sqrt{25^2-4\cdot 7^2}=\sqrt{429}\text{ cm}\doteq 20.7123\text{ cm } \\ {S}_1={\displaystyle \frac{\sqrt{3}\cdot a^2}{4}}={\displaystyle \frac{\sqrt{3}\cdot 7^2}{4}}\doteq 21.2176{\text{cm}}^2 \\ S=6\cdot S_1=6\cdot 21.2176\doteq 127.3057 \\ V=S\cdot h=127.3057\cdot 20.7123=2636.7965{\text{cm}}^3 \end{gathered}$$

Try another example

Related math problems and questions:

• 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.
• Quadrangular prism
  Calculate the volume and surface area of a regular quadrangular prism 35 cm high and the base diagonal of 22 cm.
• Four sided prism
  Calculate the volume and surface area of a regular quadrangular prism whose height is 28.6cm and the body diagonal forms a 50-degree angle with the base plane.
• Hexagonal prism 2
  The regular hexagonal prism has a surface of 140 cm² and height of 5 cm. Calculate its volume.
• Axial cut of a rectangle
  Calculate the volume and surface of the cylinder whose axial cut is a rectangle 15 cm wide with a diagonal of 25 cm long.
• Body diagonal
  Calculate the volume of a cuboid whose body diagonal u is equal to 6.1 cm. Rectangular base has dimensions of 3.2 cm and 2.4 cm
• Cuboid diagonals
  The cuboid has dimensions of 15, 20 and 40 cm. Calculate its volume and surface, the length of the body diagonal and the lengths of all three wall diagonals.
• The regular
  The regular triangular prism has a base in the shape of an isosceles triangle with a base of 86 mm and 6.4 cm arms, the height of the prism is 24 cm. Calculate its volume.
• Triangular prism,
  The regular triangular prism, whose edges are identical, has a surface of 2514 cm ^ 2 (square). Find the volume of this body in cm³ (l).
• Triangular prism - regular
  The regular triangular prism is 7 cm high. Its base is an equilateral triangle whose height is 3 cm. Calculate the surface and volume of this prism.
• Wall height
  Calculate the height of a regular hexagonal pyramid with a base edge of 5 cm and a wall height w = 20 cm.
• Diagonals of the rhombus
  How long are the diagonals e, f in the diamond, if its side is 5 cm long and its area is 20 cm²?
• Block or cuboid
  The wall diagonals of the block have sizes of √29cm, √34cm, √13cm. Calculate the surface and volume of the block.
• 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.
• Body diagonal - cube
  Calculate the surface and cube volume with body diagonal 15 cm long.
• Quadrilateral prism
  Calculate the volume (V) and the surface (S) of a regular quadrilateral prism whose height is 28.6 cm and the deviation of the body diagonal from the base plane is 50°.
• 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.