Six unit circles are arranged inside a rectangle. the circles are tangent to each other and tangent to the rectangle as they appear in the diagram. What is the area of the rectangle?
2D * 3 D = 6 D^2 = 6 (2R)^2 = 24 R^2
To find the area of the rectangle, we first need to understand the dimensions of the rectangle. Here's how you can approach this problem:
1. Start by drawing a diagram of the given setup. Make sure to label the dimensions of the rectangle and any other relevant information.
2. An important property to note is that the diameter of each unit circle is equal to the width of the rectangle. Since the circles are tangent to each other and to the rectangle, the four centers of the outer circles create a rectangle with side lengths equal to the diameters of the circles.
3. Let's assume that the diameter of each circle is d (which is equal to the width of the rectangle) and the radius of each circle is r (which is half the diameter).
4. Since each circle is tangent to its adjacent circles and to the rectangle, the centers of the unit circles will form two rows of three circles each, with the centers aligned horizontally.
5. The total height of the rectangle is equal to the diameter of one circle plus twice the radius of a circle. So, the height of the rectangle is equal to d + 2r.
6. The total width of the rectangle is equal to the diameter of one circle plus twice the diameter of another circle, since the centers of the circles are aligned horizontally. So, the width of the rectangle is equal to d + 2d = 3d.
7. Now that we have the dimensions of the rectangle (height = d + 2r and width = 3d), we can calculate the area of the rectangle.
8. The area of a rectangle is given by the formula: Area = length × width. In this case, the area is: Area = (d + 2r) × (3d).
9. Since the diameter of the unit circle is equal to 2 times the radius, we can substitute 2r for d in the formula: Area = (2r + 2r) × (3(2r)) = 4r × 6r = 24r^2.
10. Therefore, the area of the rectangle is equal to 24 times the square of the radius of the unit circle.
Please note that without specific measurements for the radius or diameter, we cannot provide an exact numerical answer.