In the figure, A, B, C, D, E, F and P, Q, R, S, T, U are 6 points dividing the each circumference of the bottom of the cylinder into six equally respectively.
Each of AP, BQ, CR, DS, ET and FU is vertical to the bottom.
The area of triangle ACE is 3 cm2 and the height of the pillar is 5 cm.
Find the volume of the solid which have two triangles ACE and QSU as the bottoms and have six triangles AQU, CSQ, EUS, ACQ, CES, and EAU as side faces.
Each of AP, BQ, CR, DS, ET and FU is vertical to the bottom.
The area of triangle ACE is 3 cm2 and the height of the pillar is 5 cm.
Find the volume of the solid which have two triangles ACE and QSU as the bottoms and have six triangles AQU, CSQ, EUS, ACQ, CES, and EAU as side faces.
Answer
Fig. 2 shows that the base area of a hexagonal pillar is twice the area of △ACE.
Since the area of △ACE is 3cm2, the area of a right hexagon is 3 × 2 = 6cm2.
Since the height of this solid is 5 cm, the volume is 6cm2 × 5cm = 30cm3.
According to Fig. 2, the base area of triangular pyramid U-AEF is 1/3 of △ACE.
It is 3cm × 1/3 = 1cm2.
The volume is 1cm2 × 5 × 1/3 = 53cm3.
The volume of these six triangular pyramids is 5/3 × 6 = 10cm3.
Therefore, the volume of the solid is 30 - 10 = 20cm3.
20 cm3
Solution
The bottom of this solid is as it is shown in Fig.1.
It is considered that the original bottom of this solid is not a circle but a right hexagon as shown in Fig. 2.
It turns out that this solid was made of cutting out six triangular pyramid U-AEF of Fig. 3 from a right hexagonal pillar.
Fig. 2 shows that the base area of a hexagonal pillar is twice the area of △ACE.
Since the area of △ACE is 3cm2, the area of a right hexagon is 3 × 2 = 6cm2.
Since the height of this solid is 5 cm, the volume is 6cm2 × 5cm = 30cm3.
According to Fig. 2, the base area of triangular pyramid U-AEF is 1/3 of △ACE.
It is 3cm × 1/3 = 1cm2.
The volume is 1cm2 × 5 × 1/3 = 53cm3.
The volume of these six triangular pyramids is 5/3 × 6 = 10cm3.
Therefore, the volume of the solid is 30 - 10 = 20cm3.
