Finding the Circumference of a Circle Inscribing a Given Rectangle
Finding the Circumference of a Circle Inscribing a Given Rectangle
When a rectangle is inscribed in a circle, the diagonal of the rectangle becomes the diameter of the circle. This relationship makes it possible to calculate the circle's circumference if we know the rectangle's dimensions. In this article, we will walk through the steps to find the circumference of the circle for a rectangle with dimensions 1.5 inches by 1 inch.
Understanding the Problem
Given a rectangle with dimensions 1.5 inches by 1 inch, we want to find the circumference of the circle that circumscribes this rectangle. The key here is to first determine the length of the diagonal of the rectangle, which acts as the diameter of the circle.
Calculating the Diagonal
To find the diagonal of the rectangle, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. For a rectangle, the diagonal forms a right-angled triangle with the two sides of the rectangle.
Draw a rectangle with dimensions 1.5 inches by 1 inch. Draw diagonals from each corner of the rectangle. Where the diagonals intersect, you have the center of the circle. Calculate the length of one of the diagonals using the Pythagorean theorem.Using the Pythagorean theorem:
d sqrt{1.5^2 1^2}
d sqrt{2.25 1} sqrt{3.25}
d approx 1.802775637731995
Determining the Radius and Circumference
The diameter of the circle is the length of the diagonal, which is approximately 1.802775637731995 inches. Next, we need to find the radius of the circle, which is half of the diameter:
r frac{d}{2} approx frac{1.802775637731995}{2} approx 0.9013878188659975
The circumference of the circle is given by the formula:
C 2pi r
Substituting the radius:
C approx 2pi times 0.9013878188659975 approx 5.654866776461628
Simplifying the Calculation
Let's simplify the calculation step-by-step:
Calculate the square of the rectangle's sides: 1.5^2 2.25 1^2 1 Add the squares: 2.25 1 3.25 Take the square root of the sum: sqrt{3.25} approx 1.80277 inches (diameter) Divide the diameter by 2 to find the radius: 0.9013878188659975 inches Multiply the radius by (2pi): 5.654866776461628) inches (circumference)The circumference of the circle, rounded to the nearest hundredth, is:
boxed{5.65} text{ inches}
Conclusion
By using the Pythagorean theorem and basic circle properties, we can easily find the circumference of a circle that circumscribes a given rectangle. In this example, the circumference of the circle is approximately 5.65 inches, providing a clear and straightforward solution to the problem.
Additional Resources
If you need further assistance with geometry or want to explore more mathematical concepts, you can find additional resources on websites such as Khan Academy, Mathway, and Math is Fun. These platforms offer a range of tutorials and practice problems to help you improve your skills.
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