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 mm³ has a rectangular base of length 10 mm and width 8 mm. Find the height h of the prism.
rectangular prism problem 1

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.
cube problem 2

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.
triangular prism problem 3

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.
L-shaped structure problem 4

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.
Hollow cylinder 5

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.
U-shaped structure problem 6

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.
hexagonal prism 7

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)