Cones Arrangement in the Box

The problem describes four identical cones placed compactly inside a box in an upright position. When identical circular bases are placed compactly side-by-side in a square pattern, they form a 2x2 arrangement within the box. Since the cones are upright, their tips are directly above the centers of their bases.

Calculating the Side Length of the Square

We are given that the base diameter of each cone is 10 cm. The radius of the base is half of the diameter.

• Base diameter = 10 cm
• Base radius ($r$) = Diameter / 2 = 10 cm / 2 = 5 cm

When the four cones are placed compactly in a square arrangement, the distance between the centers of the bases of two adjacent cones (either side-by-side in a row or front-to-back in a column) is equal to the sum of their radii. Since the cones are identical, this distance is $r + r = 2r$.

Distance between adjacent base centers = $2 \times 5$ cm = 10 cm.

Since the cones are placed upright, the tips of the cones are vertically aligned with the centers of their bases. Therefore, the horizontal distance between the tip of one cone and the tip of an adjacent cone in the arrangement is the same as the distance between their base centers.

The tips of the four cones form a square. The side length of this square is equal to the distance between the tips of any two adjacent cones.

Side length of the square formed by tips = Distance between adjacent base centers = 10 cm.

Area Calculation

The area of a square is calculated by squaring the length of its side.

Area of square = $(\text{Side length})^2$

Area = $(10 \text{ cm})^2 = 10 \text{ cm} \times 10 \text{ cm} = 100 \text{ cm}^2$

Thus, the area of the square formed by connecting the tips of the cones is 100 cm².