As a professional tutor, I will explain how to calculate the perimeter of a badge in the shape of a quarter circle with a 9 cm base and 9 cm height.
Let's break the badge down into its three segments: the two straight sides (the base and height) and the curved outer edge.
1. Straight sides: The base and height are both 9 cm, so this part is straightforward. The combined length of the base and height is 9 cm + 9 cm = 18 cm.
2. Curved outer edge: To calculate the curved outer edge, we need to understand that it is one-fourth of the circumference of the entire circle from which it is cut. The formula for the circumference C of a circle with radius r is C = 2πr, where π (pi) is a mathematical constant approximately equal to 3.14. Since the base and height are both 9 cm, it's clear that the radius of the circle is 9 cm. Thus, the circumference of the full circle is C = 2π(9 cm) ≈ 56.55 cm. Since we only have a quarter of the circle, the curved outer edge is 1/4 of this, which is about 56.55 cm / 4 ≈ 14.14 cm.
3. Perimeter: Finally, add up the lengths of the three segments: 18 cm (straight sides) + 14.14 cm (curved outer edge) ≈ 32.14 cm.
So, the perimeter of the badge in the shape of a quarter circle with a 9 cm base and 9 cm height is approximately 32.14 cm (rounded to two decimal places).