# Word problem about volumes and areas

PS = 20.8 cm. PQ = 36 cm This diagram shows 6 gift container in the shape of a regular hexagon of sides 6 cm.

The height of both hexagon gift container and the box is 8 cm .

Calculate the volume of empty space in the box which is not occupied by the gift container .

My workings -

$Vol. of box = (20.8)(36)(8) = 5990.4cm^3$

Next , I divided the hexagon into 6 equal triangles then ...

Vol of 1 hexagon =area of cross section X height = $6(1/2(6)(6)(sin60)) X 8 = 748.2459cm^3$

Vol of 6 hexgon = $748.2459 X 6 = 4489.475cm^3$

Vol of empty space = $5990.4 - 4489.4756 = 1500 cm^3 (3 sf)$

My answer is wrong ... Can I get help on why ? I calculated the total volume of the box then take away volume of 6 hexagons.

Thanks ..

• I got the same answer as you after doing the problem, so I feel like $1500$ cm$^3$ is the right answer. – Ethan Hunt Apr 9 '16 at 3:21
• Yes $1500 cm ^3$ is the right answer. I solved it another way below. – M47145 Apr 9 '16 at 17:21

Divide up each of the hexagons into six equilateral triangles. Now the entire area consists of $36+12=48$ such triangles. The $36$ are the white ones, so the empty area in black is $B = \frac{12}{48}T=0.25T$ where $T=20.8 \times 36$ is the total area. Thus $B=0.25\times 20.8 \times 36 = 187.2$
The volume of the empty part is thus $8 \times B = 1497.6 \approx 1500cm^3$.