Find Perimeter

The general way to find the perimeter of any shape is to add up the length of all its sides. For certain shapes, such as rectangles and circles, there are specific formulas you can use to simplify the process. In other instances, you might be missing one or more of the side lengths, but are given other information. In cases like this, you must complete extra steps to find the missing side length before you can calculate the perimeter.

Steps

Finding the Perimeter of Rectangles

  1. Set up the formula for the perimeter of a rectangle. The formula is <math>P = 2(w + h)</math>, where <math>P</math> equals the perimeter of the rectangle, <math>w</math> equals the width of the rectangle, and <math>h</math> equals the height of the triangle.[1] If you don’t know the length of the width and height of the rectangle, you cannot use this formula.
    • You can also use the formula <math>P = a + b +c +d</math>, where each variable is equal to the length of one side of the rectangle.
  2. Plug the width and height into the formula. Due to the commutative property, it doesn’t matter which measurement you use for the width, and which you use for the height. The width and height are two adjacent sides. If the rectangle is not a square, these to side lengths must be different.
    • For example, if a rectangle has a width of 5 cm and a height of 10 cm, your formula will look like this: <math>P = 2(5 + 10)</math>.
  3. Add the length and width, and multiply by 2. Make sure you follow the order of operations and complete the calculation in parentheses before multiplying. The resulting value will give you the perimeter of your rectangle.
    • For example:
      <math>P = 2(5 + 10)</math>
      <math>P = 2(15)</math>
      <math>P = 30</math>
      So, the perimeter of the rectangle is 30 cm.
  4. Use the formula <math>P = 4x</math> to find the perimeter of a square. In this formula <math>x</math> is equal to the length of one side of the square. A square has 4 equal sides, so to find its perimeter, you only need to multiply the length of one side by 4.[2]
    • For example, if a square has one side that is 3 cm long, to find the perimeter, you would calculate <math>P = 4(3) = 12</math>. So, the perimeter is 12 cm.
  5. Find the perimeter given other information. Often you will not be given the length of all sides, or even the length of any side. It still may be possible to find the perimeter of a rectangle.
    • If you know the area of the rectangle, and the length of one side, you can find the perimeter by finding the missing width or height using the area formula. Set up the formula <math>A = wh</math>.[3] Plug in the values you know, then solve for the missing variable. Now you know the length and width, so you can use the perimeter formula.
    • If you know one side length and the length of the diagonal, you can use the Pythagorean Theorem to find the missing side length. Set up the formula <math>a^{2} + b^{2} = c^{2}</math>. Substitute the length of the diagonal for <math>c</math>, and the side length for <math>a</math>. Solve for <math>b</math>. Now you know the length and width, so you can use the perimeter formula.[4]

Finding the Perimeter of a Circle

  1. Set up the formula for finding the circumference of a circle. The circumference is the distance around the circle, and is thus the same as its perimeter. The formula is <math>C = 2\pi \cdot r</math>, where <math>C</math> equals the circumference and <math>r</math> equals the radius. Since the radius is half the diameter, you can use the formula <math>C = \pi(d)</math> if you have the diameter instead of the radius.[5]
  2. Plug the length of the radius into the formula. Make sure you substitute for the variable <math>r</math>. If you are using the diameter formula, substitute for <math>d</math>. The length of the radius or diameter should be given, or you should be able to measure it. If you do not have this information, you can’t use these formulas.
    • For example, if the radius of the circle is 6 cm, your formula will look like this:<math>C = 2\pi \cdot 6</math>.
  3. Multiply the radius by <math>2\pi</math>. You can use 3.14 for <math>\pi</math>, but if you are using a calculator you can use the <math>\pi</math> key for a more precise answer. The product of these three values is equal to the circumference, or perimeter, of the circle.
    • For example: <math>C = 2\pi \cdot 6 = 37.7</math>. So the circumference of the circle is 37.7 cm.
  4. Find the perimeter given the area. The area of a circle is given by the formula <math>A = \pi \cdot r^{2}</math>. So, if you plug the area into the formula, you can solve for <math>r</math>. Once you have <math>r</math>, you can use the circumference formula to find the circumference.[6]
    • For example, if you are told that the area of a circle is 64 square centimeters, you would set up the formula <math>64 = \pi \cdot r^{2}</math>. Then, use the Do Algebra to solve for <math>r</math>:
      <math>64 = \pi \cdot r^{2}</math>
      <math>\frac{64}{\pi} = \frac{\pi \cdot r^{2}}{\pi}</math>
      <math>20.37 = r^{2}</math>
      <math>\sqrt{20.37} = \sqrt{r^{2}}</math>
      <math>4.51 = r</math>
      So, the radius of the circle is about 4.51 cm. Now you can plug this value into the perimeter formula and solve.

Finding the Perimeter of Triangles

  1. Set up the formula for finding the perimeter of a triangle. The formula is <math>P = a + b + c</math>, where the variables equal the three sides of the triangle. This formula is the same whether or not the triangle is right. You must have all side lengths to use this formula. If you know that you have an equilateral triangle, you only need one side length, since an equilateral triangle has three equal sides.[7]
    • For example, if a triangle has sides that are 5, 7, and 12 cm in length, you simply add up all the side lengths to find the perimeter: <math>P = 5 + 7 + 12 = 24</math>. So, the perimeter of the triangle is 24 cm.
  2. Find the perimeter of a right triangle with a missing side length. Sometimes you might be presented with a right triangle that only has two side lengths given. In this case, set up the Pythagorean formula to find the missing side length. The formula is <math>a^{2} + b^{2} = c^{2}</math>, where <math>c</math> is the length of the hypotenuse (the side opposite the right angle), and <math>a</math> and <math>b</math> are the other two side lengths. Solve for the missing variable, and this will give you your missing side length.[8]
    • For example, if you have a right triangle with a hypotenuse of 10 cm, and one side length of 6 cm, set up the Pythagorean formula like this: <math>6^{2} + b^{2} = 10^{2}</math>
    • Solve for <math>b</math>:
      <math>36 + b^{2} = 100</math>
      <math>36 + b^{2} - 36 = 100 - 36</math>
      <math>b^{2} = 64</math>
      <math>\sqrt{b^{2}} = \sqrt{64}</math>
      <math>b = 8</math>
    • Now that you have all three side lengths, you can add them up to find the perimeter: <math>10 + 6 + 8 = 24</math>. So, the perimeter of the triangle is 24 cm.
  3. Find the perimeter of an isosceles triangle with a missing side length. Since the height, or altitude, of an isosceles triangle bisects the base, if you know the height and base of the triangle, you can use the Pythagorean theorem to find the missing side lengths.[9]
    • For example, if an isosceles triangle has a height of 10 cm and a base of 6 cm, you can think of the height creating two right triangles. Since the height bisects the base, one side length of the right triangle will be 3 cm. The other side length will be equal to the height: 10 cm. The missing side length is the hypotenuse.
    • Set up the Pythagorean formula, plugging in the side lengths: <math>10^{2} + 3^{2} = c^{2}</math>.
    • Make the necessary calculations to find the missing side length:
      <math>100 + 9 = c^{2}</math>
      <math>109 = c^{2}</math>
      <math>\sqrt{109} = \sqrt{c^{2}}</math>
      <math>10.44 = c</math>.
    • Remember that an isosceles triangle has two equal sides. So, the perimeter of the triangle is equal to <math>2x + b</math>, where <math>x</math> equals the length of one side, and <math>b</math> equals the base. So, if you know the length of the base and one side, you can find the perimeter of an isosceles triangle: <math>P = 2(10.44) + 6 = 26.88</math>. So, the perimeter of the triangle is 26.88 cm.

Finding the Perimeter of a Regular Polygon

  1. Find the length of one side. This information might be given to you. If it isn’t, you can find the length of one side if you know the length of the polygon’s apothem or radius. The apothem is the distance between the center of the polygon to the midpoint of any side, and the radius is the distance between the center of the polygon and any vertex.
    • To find a side length given the apothem, use the formula <math>x = 2A \text{tan}(\frac{180}{n})</math>, where <math>x</math> equals the side length and <math>A</math> equals the apothem.[10]
    • To find the side length given the radius, use the formula <math>x = 2r \text{sin}(\frac{180}{n})</math>, where <math>x</math> equals the side length and <math>r</math> equals the radius.[10]
    • For example, if the radius of a hexagon is 5 cm, to find the side length, you would calculate:
      <math>x = 2(5) \text{sin}(\frac{180}{6})</math>
      <math>x = 2(5) \text{sin}(30)</math>
      <math>x = 2(5)(.5)</math>
      <math>x = 5</math>
  2. Set up the formula for the perimeter of a regular polygon. The formula is <math>P = nx</math>, where <math>n</math> is the number of sides the polygon has, and <math>x</math> is the length of one side.[11]
  3. Plug the values of <math>x</math> and <math>n</math> into the formula. Multiply these two values to find the perimeter of the polygon.
    • For example, if a regular hexagon has a side length of 5 cm, you would calculate <math>P = (6)(5) = 30</math>. So, the perimeter of the hexagon is 30 cm.

Tips

  1. To find the perimeter of a trapezoid when you are missing side lengths, in general you want to divide the trapezoid into two right triangles and one rectangle. From there you can use the properties of right triangles and rectangles to find the missing side lengths.
  2. To Find the Perimeter of a Rhombus when you are missing side lengths, in general you want to use the diagonal(s) of the rhombus to divide the shape into several right triangles. Then you can use the Pythagorean theorem or trigonometry to find the missing side lengths.



Related Articles

Sources and Citations

  1. ↑ http://www.mathopenref.com/rectangleperimeter.html
  2. ↑ https://www.mathsisfun.com/geometry/square.html
  3. ↑ http://www.virtualnerd.com/pre-algebra/perimeter-area-volume/perimeter-and-area/area-formulas-examples/rectangle-area-example
  4. ↑ https://www.youtube.com/watch?v=EIWGr_NcnJA
  5. ↑ http://www.mathplanet.com/education/pre-algebra/more-about-equation-and-inequalities/calculating-the-circumference-of-a-circle
  6. ↑ https://learnzillion.com/lesson_plans/9072-use-area-of-a-circle-to-solve-for-circumference
  7. ↑ http://mathworld.wolfram.com/EquilateralTriangle.html
  8. ↑ http://www.varsitytutors.com/basic_geometry-help/how-to-find-the-perimeter-of-a-right-triangle
  9. ↑ http://www.mathopenref.com/isosceles.html
  10. ↑ 10.0 10.1 http://www.mathopenref.com/polygonsides.html
  11. ↑ http://www.mathopenref.com/polygonperimeter.html
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