Regular square prism

The volume of a regular square prism is 192 cm³. The size of its base edge and the body height is 1: 3. Calculate the surface of the prism.

Correct answer:

S =  218.9202 cm²

Step-by-step explanation:

$$\begin{gathered} V=192{\text{cm}}^3 \\ a:d=1:3 \\ d=3a \\ V=a^2\cdot h \\ s=\sqrt{2}\cdot a \\ {h}^2=d^2-s^2=d^2-2a^2=9a^2-2a^2 \\ {h}^2=7a^2 \\ h=\sqrt{7}a \\ V=a^2\cdot h \\ V=a^2\cdot \sqrt{7}a \\ V=a^3\cdot \sqrt{7} \\ a=\sqrt[3]{V\mathrm{/}\sqrt{7}}=\sqrt[3]{192\mathrm{/}\sqrt{7}}\doteq 4.1711\text{ cm } \\ h=\sqrt{7}\cdot a=\sqrt{7}\cdot 4.1711\doteq 11.0357\text{ cm } \\ S=2\cdot a^2+4\cdot a\cdot h=2\cdot 4.1711^2+4\cdot 4.1711\cdot 11.0357\doteq 218.9202{\text{cm}}^2 \\ \text{ Verifying Solution: } \\ {V}_2=a^2\cdot h=4.1711^2\cdot 11.0357=192{\text{cm}}^3 \\ {d}_2=\sqrt{h^2+a^2+a^2}=\sqrt{11.0357^2+4.1711^2+4.1711^2}\doteq 12.5133\text{ cm } \\ {k}_2=d_2\mathrm{/}a=12.5133\mathrm{/}4.1711=3 \end{gathered}$$

Try another example

Related math problems and questions:

• Body diagonal
  The cuboid has a volume of 32 cm³. Its side surface area is double as one of the square bases. What is the length of the body diagonal?
• 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
• Ratio-cuboid
  The lengths of the edges of the cuboid are in the ratio 2: 3: 6. Its body diagonal is 14 cm long. Calculate the volume and surface area of the cuboid.
• 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.
• Cuboid walls
  Calculate the cuboid volume if its different walls have an area of 195cm², 135cm², and 117cm².
• Quadrangular prism
  Calculate the volume and surface area of a regular quadrangular prism 35 cm high and the base diagonal of 22 cm.
• 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.
• 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°.
• Cube diagonals
  Determine the volume and surface area of the cube if you know the length of the body diagonal u = 216 cm.
• Body diagonal
  Cuboid with base 7cm x 3,9cm and body diagonal 9cm long. Find the height of the cuboid and the length of the diagonal of the base,
• Rhombus base
  Calculate the volume and surface area of prisms whose base is a rhombus with diagonals u₁ = 12 cm and u₂ = 15 cm. Prism height is twice the base edge length.
• Quadrilateral prism
  The height of a regular quadrilateral prism is v = 10 cm, the deviation of the body diagonal from the base is 60°. Determine the length of the base edges, the surface, and the volume of the prism.
• Equilateral cylinder
  Equilateral cylinder (height = base diameter; h = 2r) has a volume of V = 199 cm³ . Calculate the surface area of the cylinder.
• Rotary cylinder
  In the rotary cylinder it is given: surface S = 96 cm² and volume V = 192 cm cubic. Calculate its radius and height.
• Nice prism
  Calculate the cuboid's surface if the sum of its edges is a + b + c = 19 cm and the body diagonal size u = 13 cm.
• Body diagonal - cube
  Calculate the surface and cube volume with body diagonal 15 cm long.
• 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.