Determine a Square and Circle of Equal Perimeter

A perimeter is generally thought of as the sum of all the side lengths of a shape. Like all two-dimensional shapes, a circle has a perimeter (usually called the circumference), but since a circle does not have sides its perimeter is calculated differently than that of other polygons. Although one has sides and the other is a curved line, it is possible to have a square and a circle with the same perimeter. Given either the radius of the circle or the side length of the square, you can determine the dimensions of both shapes with an equal perimeter.

Steps

  1. Recognize the basic relations you'll use. The perimeter of square, with side <math>s</math>, is <math>4s</math>. The circumference of a circle is <math>2\pi(r)</math> where <math>r</math> is the radius. Therefore, you're dealing with <math>4s = 2\pi(r)</math>, and you can use these equations to help find your side or radius: <math>s =\frac{\pi(r)}{2}</math> and <math>r = \frac{2(s)}{\pi}</math>, depending on which method you use, below.

Given the Radius of a Circle

  1. Set up the formula for the perimeter, or circumference, of a circle. The formula is <math>2\pi(r)</math>, where <math>r</math> equals the length of the radius.[1]
    • The radius is the measurement between the center of a circle and its edge.[2]
  2. Plug the value of <math>r</math> into the formula. This information should be given, or you should be able to measure the radius. If you do not know the length of the radius, you cannot use this method.
    • For example, the circumference of a circle with a radius of 4cm is shown by <math>2\pi4</math>cm.
  3. Find the circumference of the circle. To do this, multiply the three values together. If you are not using a calculator, substitute 3.14 for the value of <math>\pi</math>.
    • For example, <math>2 \times 3.14 \times 4 = 25.12</math>. So the circumference, or perimeter, of the circle is 25.12cm.
  4. Set up the formula for the perimeter of a square. The formula is <math>P = 4s</math>, where <math>P</math> equals the perimeter of the square and <math>s</math> equals the side length of the square.[3]
  5. Plug the circumference of the circle into the formula. You are substituting the circumference for <math>P</math>, since the circumference of the circle and the perimeter of the square are supposed to be equal.
    • For example, <math>25.12 = 4s</math>.
  6. Solve for <math>s</math>. To do this, divide both sides of the equation by 4. This will give you the side length of the square.
    • For example:
      <math>25.12 = 4s</math>
      <math>\frac{25.12}{4} = \frac{4s}{4}</math>
      <math>6.28 = s</math>.
      So the circumference of a circle with a radius of 4 cm is equal to the perimeter of a square with side lengths of 6.28cm.

Given the Side Length of a Square

  1. Set up the formula for the perimeter of a square. The formula is <math>4s</math>, where <math>s</math> equals the side length of the square.[3]
    • A square has four equal sides, so you only need to know the length of one side to find the perimeter.
  2. Plug the value of <math>s</math> into the formula. This information should be given, or you should be able to measure the side. If you do not know the length of the side, you cannot use this method.
    • For example, the perimeter of a square with a side length of cm is shown by <math>4(4)</math>.
  3. Find the perimeter of the square. To do this, multiply the side length by 4.
    • For example, <math>4 \times 4 = 16</math>. So the perimeter of the square is 16 cm.
  4. Set up the formula for the perimeter, or circumference, of a circle. The formula is <math>C = 2\pi(r)</math>, where <math>C</math> equals the circumference of the circle and <math>r</math> equals the length of the radius.[1]
    • The radius is the measurement between a circle’s center and edge.[2]
  5. Plug the perimeter of the square into the formula. You are substituting the perimeter for <math>C</math>, since the perimeter of the circle and the circumference of the circle are supposed to be equal.
    • For example, <math>16 = 2\pi(r)</math>.
  6. Solve for <math>r</math>. To do this, divide each side by <math>2\pi</math>. If you are not using a calculator, substitute 3.14 for the value of <math>\pi</math>. This will give you the radius of the circle.
    • For example:
      <math>16 = 2\pi(r)</math>
      <math>\frac{16}{2\pi} = \frac{2\pi(r)}{2\pi}</math>
      <math>\frac{16}{6.28} = \frac{6.28r}{6.28\pi}</math>
      <math>2.55 = r</math>
      So the perimeter of a square with a side length of of 4cm is equal to the circumference of a circle with a radius of 2.55cm.

Tips

  • You can find a square and circle with the same perimeter if given the area of either shape. The formula for the area of a circle is <math>\pi(r^{2})</math>. The formula for the area of a square is <math>s^{2}</math>. Solve for <math>r</math> or <math>s</math> first, then follow the steps above.

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Sources and Citations

  1. ↑ 1.0 1.1 http://www.mathopenref.com/circumference.html
  2. ↑ 2.0 2.1 https://www.mathsisfun.com/definitions/radius.html
  3. ↑ 3.0 3.1 http://www.purplemath.com/modules/geoform.htm
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