3D shapes, such as prisms, volume problems 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.
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
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
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)