3D Shapes Volume Problems
Problems on 3D shapes, such as prisms, cube, cylinder, volume are presented along with detailed solutions
Problem 1
A 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 2
The 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 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 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 4
Find the volume of the given Lshaped 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 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 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 6
Find x so that the volume of the Ushaped 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 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 (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.
