Problems on 3D shapes, such as prisms, cube, cylinder, volume are presented along with detailed solutions

## Problem 1A rectangular prism of volume 3200 mm^{3} has a rectangular base of length 10 mm and width 8 mm. Find the height h of the prism.
Solution to Problem 1:
- Volume is given by by
volume = length * width * height = 10 mm * 8 mm * h = 3200 mm^{3}
- Solve for h
h = 3200 mm^{3}/ 80 mm^{2}= 40 mm
## Problem 2The area of one square face of a cube is equal to 64 cm^{2}. Find the volume of the cube.
Solution to Problem 2:
- The area of one square face is given by
s * s = 64 cm^{2}
- Solve for s
s = SQRT(64 cm^{2}) = 8 cm
- The volume V of the given cube is given by
V = s^{3}= 8^{3}= 512 cm^{3}
## Problem 3The 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 mm^{3}, 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 mm^{3}
- Substitute b by 2a and h by its value
40 mm^{3}= a^{2}* 10 mm
- Solve for a and calculate b
a = 2 mm b = 2a = 4 mm
## Problem 4Find 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 mm^{3}- 40 * 60 * 10 mm^{3}= 24000 mm^{3}
## Problem 5Find 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 mm^{3} 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 R^{2}- Pi r^{2}) = 11000 Pi
- Also R = 6 and h = 100 cm = 1000 mm, hence
1000 * (36 Pi - Pi r^{2}) = 11000 Pi
- Solve for r
r = 5 mm
- Find x
x = R - r = 1 mm
## Problem 6Find x so that the volume of the U-shaped rectangular structure is equal to 165 cm^{3}.
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 mm^{3}- x * x * 3 mm^{3}= 165 cm^{3}
- Solve for x
x = 5 cm
## Problem 7Find 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 (x^{2}SQRT(3) / 4)
- Hence the volume V of the prism
V = 24 * 6 (10^{2}SQRT(3) / 4) cm^{2}= 6235.4 cm^{3}(rounded to 1 decimal place)
## More References and Links to Triangles and GeometryGeometry Tutorials, Problems and Interactive Applets. |