Answer:
11700 mm²
Explanation:
You want the area of the rectangle enclosing three packed circles with radius 28 mm.
Width
The width of the rectangle is 4 times the radius of a circle, so is ...
W = 4·(28 mm) = 112 mm
Height
The height of the rectangle is the height of an equilateral triangle with sides equal to 2 times the radius, together with 2 times the radius.
Connecting the centers of the circles makes an equilateral triangle with side length 2·28 = 56 mm. A line straight up from the center of the bottom circle divides that triangle into two congruent 30°-60°-90° right triangles.
You may already know that the side lengths of that shape triangle have the ratios 1 : √3 : 2. If so, you recognize that these lengths in this figure are 28 : 28√3 : 56 mm, so the height of the equilateral triangle is 28√3 mm.
The height of the rectangle is ...
28√3 +2·28 = 28(2+√3) ≈ 104.497 . . . . millimeters
Pythagorean theorem
In the event you don't recall the side ratios of this special triangle, you can find the height using the Pythagorean theorem. The height (h) will satisfy the relation ...
28² +h² = 56²
h² = (2·28)² -28² = 28²(4 -1) = 28²·3
h = √(28²·3) = 28√3
Area
Then the area of the rectangle is ...
(112 mm)(104.497 mm) ≈ 11700 mm² . . . . . . rounded to 3 sf
<95141404393>