Perimeter

Perimeter is the measure around the outside of a two-dimensional object.  Sometimes it’s best to draw pictures to solve perimeter problems.  When you imagine "measuring" perimeter, run your finger all the way around the outside edge of the object.  Make sure to include the length of every side that your fingers runs over when you add up the sides (and do not include any interior lines that your finger did not run over!).

• Perimeter: The distance around the outside of an object.  To find perimeter, add up the lengths of all of the sides of a polygon.
• Equilateral: All sides are equal.
• Regular: All sides are equal.  In math we usually talk about equilateral triangles (three equal sides) and regular polygons (so, a regular Octagon has 8 sides of the same length).

When a problem gives you sides, you can add them up to find perimeter.  (When students have trouble remembering what perimeter means, we often think of it as the distance your finger traces around a shape or the distance an ant would have to walk to cover the entire outer edge.)

The most basic perimeter problems will give you the dimensions of the shape and ask for the perimeter.  So, for the figure above, a problem might tell you: the length = 12 and the width = 2. Perimeter of a rectangle = length + length + width + width $$\eqalign{\text{Perimeter}&= l + l +w + w\\P &= 12 +12 + 2 + 2\\P &= 28}$$	Triangles, unlike rectangles, may or may not have equal sides.  To find the perimeter of a triangle, you just add the lengths of the sides, whatever they are.  A problem might give you: a triangle has 2 sides of length 3 and one side of length 4.  Perimeter = side + side + side $$\eqalign{\text{Perimeter}= 3 + 3 + 4\\P=10}$$

Sometimes perimeter problems are a bit more complicated than just adding up the sides.  One twist that tests like is to give the perimeter and ask the student to find the sides. 

If a problem tells you that the perimeter of a rectangle is 12 and that the width is 2, you can work backwards to find the length because you know that the perimeter is just all of the sides added together. Perimeter of a rectangle = length + length + width + width $$\eqalign{P&= l + l +w + w\\12 &= l + l + 2 + 2\\12 &= l + l + 4\\-4&\;\;-4\\8 &= 2l\\4 &= l = length}$$	If a problem tells you that the perimeter of an equilateral triangle is 18, you can work backwards to find the length of each side. If you know a triangle is equilateral (equal on all sides) you can divide the perimeter by 3 to get the length of the sides. Perimeter of equilateral triangle ÷ 3 = side. $18\div 3=6$ If the perimeter of the above triangle is 18, each side is 6 long.

Some problems are even more complicated. But the basic rules apply:  Perimeter is the sum of all the sides.  Draw the shapes (this really helps!), find the perimeters, and solve for the missing information.  It may take a few steps.  Just keep going step by step until you get your variable.

Example: The two triangles below have the same perimeter.  What is the value of $x$?

First, find the perimeter of one triangle. Start with the easier one!

Let's take the equilateral triangle, where we have all the sides given:

$$\eqalign{P&=9+9+9\\P&=27}$$

The problem said that the perimeters were equal.  Now that you know the perimeter of one, you can substitute that in for the perimeter of the other to find the missing variable: 

$$\eqalign{P&=8+9+x\\P&=17+x\\27&=17+x&&\text{Substitute 27 in for Perimeter}\\-17&\;\;-17\\10&=x}$$

The missing side, $x$, = 10.

Perimeter is a very simple idea: the distance around a shape.  Solving problems dealing with perimeter often involve breaking perimeter into several steps and working carefully towards a variable.  Just follow the rules of algebra and you'll get there.