If you're working with geometry problems involving squares and circles, understanding how to relate their measurements is key. In this example, we are given the area of a square and asked to find the radius of a circle perfectly inscribed within it. Follow the step-by-step solution below to learn how to solve such problems using basic algebra and geometry concepts.
Question: (Grade 12 General Mathematics Examination Question 41: 2011)
The diagram shows a circle inscribed in a square.
What is the radius of the circle if the area of the square is \(625 \, \text{cm}^2\)?
Solution:
Step 1: Understand the relationship
Since the circle is inscribed in the square, the diameter of the circle is equal to the side length of the square.
Step 2: Find the side length of the square
We are given:
\[
\text{Area of square} = \text{side}^2
\]
\[
625 = \text{side}^2
\]
Take the square root of both sides:
\[
\text{side} = \sqrt{625} = 25 \, \text{cm}
\]
Step 3: Find the diameter of the circle
Since the circle fits exactly inside the square:
\[
\text{Diameter of circle} = \text{side of square} = 25 \, \text{cm}
\]
Step 4: Find the radius of the circle
\[
\text{Radius} = \frac{\text{Diameter}}{2} = \frac{25}{2} = 12.5 \, \text{cm}
\]
Final Answer:
\[ \boxed{12.5 \, \text{cm}} \] Also check out