Math Problem Statement
The side of a square is 10 cm long. What is the circumference of the circle when a circle is circumscribed around the square?
Solution
To find the circumference of a circle circumscribed around a square, we need to first determine the radius of the circle. A circle circumscribed around a square will have its diameter equal to the diagonal of the square.
Steps:
-
Find the diagonal of the square:
The diagonal of a square can be calculated using the Pythagorean theorem, since the diagonal forms the hypotenuse of a right triangle where the two sides are the side lengths of the square.
For a square with side length : -
Determine the radius:
The radius of the circumscribed circle is half the diagonal: -
Find the circumference of the circle:
The circumference of a circle is given by the formula:Substituting the value of :
So, the circumference of the circumscribed circle is , or approximately 44.43 cm when evaluated numerically.
Do you want further details or have any questions about this?
Here are some related questions:
- What is the area of the square in this scenario?
- How would you calculate the area of the circumscribed circle?
- What is the length of the diagonal of the square in decimal form?
- Can you find the perimeter of the square and compare it with the circle’s circumference?
- How would you compute the area difference between the square and the circle?
Tip: When dealing with a circumscribed circle around a square, always start by finding the diagonal of the square, as it becomes the diameter of the circle.
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Math Problem Analysis
Mathematical Concepts
Geometry
Circles
Squares
Pythagorean Theorem
Formulas
Diagonal of square: d = √(s^2 + s^2)
Radius of circumscribed circle: r = d / 2
Circumference of a circle: C = 2πr
Theorems
Pythagorean Theorem
Suitable Grade Level
Grades 8-10
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