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:

  1. Find the diagonal of the square:
    The diagonal dd 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 s=10cms = 10 \, \text{cm}:

    d=s2+s2=102+102=100+100=200=102cmd = \sqrt{s^2 + s^2} = \sqrt{10^2 + 10^2} = \sqrt{100 + 100} = \sqrt{200} = 10\sqrt{2} \, \text{cm}

  2. Determine the radius:
    The radius rr of the circumscribed circle is half the diagonal:

    r=d2=1022=52cmr = \frac{d}{2} = \frac{10\sqrt{2}}{2} = 5\sqrt{2} \, \text{cm}

  3. Find the circumference of the circle:
    The circumference CC of a circle is given by the formula:

    C=2πrC = 2\pi r

    Substituting the value of r=52r = 5\sqrt{2}:

    C=2π×52=10π2cmC = 2\pi \times 5\sqrt{2} = 10\pi\sqrt{2} \, \text{cm}

So, the circumference of the circumscribed circle is 10π2cm10\pi\sqrt{2} \, \text{cm}, or approximately 44.43 cm when evaluated numerically.


Do you want further details or have any questions about this?

Here are some related questions:

  1. What is the area of the square in this scenario?
  2. How would you calculate the area of the circumscribed circle?
  3. What is the length of the diagonal of the square in decimal form?
  4. Can you find the perimeter of the square and compare it with the circle’s circumference?
  5. 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