Volume of a Sopd made of Cubes with Unit Fraction Edge Lengths

Here we find volume of sopds made of cubes with unit fraction edge lengths. Consider for example a sopd of dimensions 3 in × 3 in × 3 made of small cubes with $frac{1}{2}$ inch edge lengths.

In that case the sopd is made up of 6 × 6 × 6 small cubes of $frac{1}{2}$ inch edge lengths. So the volume of the sopd in this case would be

Volume = l w h = $6 imes frac{1}{2} imes 6 imes frac{1}{2} imes 6 imes frac{1}{2}$

= 3 × 3 × 3 = 27 cubic inches

Formula for the volume of sopd made of cubes with unit fractional edge lengths

Assuming the sopd to be a cube of edge a units

b = number of cubes with unit fractional edge length along each edge

k = unit fractional edge length

Volume of sopd = b × k × b × k × b × k cubic units

Find the volume of following sopd of cubes with unit fraction edge lengths. Each prisms unit is measured in cm (not to scale)

Solution

Step 1:

Sopd of cubes with unit fraction edge lengths of $frac{1}{2}$ cm

Step 2:

Volume V = l w h = $2 frac{1}{2} imes 2 frac{1}{2} imes 2 frac{1}{2}$

= $5 imes frac{1}{2} imes 5 imes frac{1}{2} imes 5 imes frac{1}{2}$

= $15 frac{5}{8}$ cu cm

Find the volume of following sopd of cubes with unit fraction edge lengths. Each prisms unit is measured in cm (not to scale)

Solution

Step 1:

Sopd of cubes with unit fraction edge lengths of $frac{1}{3}$ cm

Step 2:

Volume V = l w h = $4 frac{1}{3} imes 4 frac{1}{3} imes 4 frac{1}{3}$

= $13 imes frac{1}{3} imes 13 imes frac{1}{3} imes 13 imes frac{1}{3}$

= $81 frac{10}{27}$ cu cm