Solution to Problem 1:

• Volume is given by by
  
  volume = length * width * height = 10 mm * 8 mm * h = 3200 mm³
  
  
• Solve for h
  
  h = 3200 mm³ / 80 mm² = 40 mm
  
  

Problem 2: The area of one square face of a cube is equal to 64 cm². Find the volume of the cube.

Solution to Problem 2:

• The area of one square face is given by
  
  s * s = 64 cm²
  
  
• Solve for s
  
  s = SQRT(64 cm²) = 8 cm
  
  
• The volume V of the given cube is given by
  
  V = s³ = 8³ = 512 cm³
  
  

Problem 3: The triangular base of a prism is a right triangle of sides a and b = 2a. The height h of the prism is equal to 10 mm and its volume is equal to 40 mm³, find the lengths of the sides a and b of the triangle.

Solution to Problem 3:

• The volume V of the prism is given by
  
  V = (1/2) a * b * h = 40 mm³
  
  
• Substitute b by 2a and h by its value
  
  40 mm³ = a² * 10 mm
  
  
• Solve for a and calculate b
  
  a = 2 mm
  b = 2a = 4 mm
  

Problem 4: Find the volume of the given L-shaped rectangular structure.

Solution to Problem 4:

• We can think of the given shape as a larger rectangular prism of dimensions 60, 80 and 10 mm from which a smaller prism of dimensions 40, 60 and 10 mm has been cut. Hence the volume V of the given 3D shape
  
  V = 60 * 80 * 10 mm³ - 40 * 60 * 10 mm³ = 24000 mm³
  
  

Problem 5: Find the thickness x of the hollow cylinder of height 100 cm if the volume between the inner and outer cylinders is equal to 11000 Pi mm³ and the outer diameter is 12 mm.

Solution to Problem 5:

• If R and r are the outer and inner radii of the hollow cylinder the volume V between the inner and outer cylinders is given by
  
  V = h*(Pi R² - Pi r²) = 11000 Pi
  
  
• Also R = 6 and h = 100 cm = 1000 mm, hence
  
  1000 * (36 Pi - Pi r²) = 11000 Pi
  
  
• Solve for r
  
  r = 5 mm
  
  
• Find x
  
  x = R - r = 1 mm
  
  

Problem 6: Find x so that the volume of the U-shaped rectangular structure is equal to 165 cm³.

Solution to Problem 6:

• We can think of the given shape as a larger rectangular prism of dimensons 8, 3 and 10 cm from which a smaller prism of dimensions x, x and 3 cm has been cut. Hence the volume V of the given 3D shape is given by
  
  V = 8 * 3 * 10 mm³ - x * x * 3 mm³ = 165 cm³
  
  
• Solve for x
  
  x = 5 cm
  

Problem 7: Find the volume of the hexagonal prism whose base is a regular hexagon of side x = 10 cm.

Solution to Problem 7:

• The hexagon is made up of 6 equilateral triangles, hence the area A of the base
  
  A = 6 (x² SQRT(3) / 4)
  
  
• Hence the volume V of the prism
  
  V = 24 * 6 (10² SQRT(3) / 4) cm ² = 6235.4 cm³ (rounded to 1 decimal place)

More references to triangles and geometry.