The diagram shows a circle inscribed in a square with a diagonal drawn.

Question

A diagram.Short description, A diagram.,Long description,
The diagram shows a circle inside a square. The inscribed circle touches each side of the square. The radius of the circle, shown as a line segment running from the center of the circle to the edge of the circle, is 6 centimeters. A diagonal connects the upper right and lower left corners of the square, dividing the square into 2 triangles.



The radius of the circle is 6 centimeters.

Question
What is the length, in centimeters, of the diagonal of the square? Round the answer to the nearest tenth. Enter the answer in the box.

Response area with 1 text input box
centimeters

Answer 1

To find the length of the diagonal of the square, we first note that the radius of the inscribed circle is 6 centimeters. Since the circle is inscribed in the square, the diameter of the circle equals the length of a side of the square.

The diameter of the circle is: \[ \text{Diameter} = 2 \times \text{Radius} = 2 \times 6 = 12 \text{ cm} \]

Let \( s \) be the side length of the square, so \( s = 12 \) cm.

To find the length of the diagonal \( d \) of a square, we use the formula: \[ d = s\sqrt{2} \]

Substituting \( s \) with 12 cm: \[ d = 12\sqrt{2} \]

Now calculate \( d \): \[ d \approx 12 \times 1.414 \approx 16.97 \text{ cm} \]

Rounding to the nearest tenth: \[ d \approx 17.0 \text{ cm} \]

Thus, the length of the diagonal of the square, rounded to the nearest tenth, is: \[ \boxed{17.0} \]