Question

Three right circular cylinders,each of height 20cm and radius 6 cm touch each other.find the volume of the region bounded by them

Answer 1

We first have one cylinder touching the other two.

We join the centers of the circles of their bases to form an equilateral triangle of side 2r.

The side of the equilateral triangle is :

2 × 6 = 12 cm

We can find the height of the triangle as follows:

12² - 6² = 108

h = √108 = 10.39

Area of the triangle is :

1/2 × 12 × 10.39 = 62.34

Within this triangle there are three sectors of each of the circles.

The area of the sectors is :

3 × 60/360 × 3.142 × 6² = 56.556

The area bound by the three cylinders is :

62.34 - 56.556 = 5.784 cm²

The volume bound by the three cylinders is :

5.784 × 20 = 115.68 cm³

Answer 2

The side of the equilateral triangle is :

2 × 6 = 12 cm

We can find the height of the triangle as follows:

12² - 6² = 108

h = √108 = 10.39

Area of the triangle is :

1/2 × 12 × 10.39 = 62.34

Within this triangle there are three sectors of each of the circles.

The area of the sectors is :

3 × 60/360 × 3.142 × 6² = 56.556

The area bound by the three cylinders is :

62.34 - 56.556 = 5.784 cm²

The volume bound by the three cylinders is :

5.784 × 20 = 115.68 cm³