Sometimes you need to figure out a shape's perimeter whether you have all the information you want or not. In this lesson, you'll learn how to find the perimeter of any quadrilateral as well as more complicated irregular and combined shapes.
Finding the Perimeter
Let's say you like running. Normally, you run on a track. You know the track is 400 meters long. If you think about it, that 400 meters is the perimeter of the oval. Perimeter, as you'll recall, is the path around the outside of a shape.
But, what if you get tired of running in endless ovals? Maybe you'd rather run around the block where you live. Or, maybe you want to wind a bit through your neighborhood. How do you know how far you went? There's where knowing how to find a perimeter is essential.
Let's start with you running around your block and wanting to know the distance, or perimeter. If you look at your block on a map, it has four sides. That makes it a quadrilateral. A quadrilateral is just a four-sided shape. If you're lucky, you know how long each side is.
North Street and South Street are each 250 meters long. East and West Streets are each 100 meters long. To find the perimeter of this block, just add up the sides. That's 250 + 250 + 100 + 100 = 700 meters. That's almost two times the length of the track, and you're already home when you're done!
But, what if you don't know how long each street is? Let's say you're staying at your cousin's house and want to run in that neighborhood. She tells you that she knows George Street is 100 meters and John Street is 50 meters. What about Paul and Ringo? Fortunately, you recognize that the corners on this block are right angles. Therefore, this is no ordinary quadrilateral; it's a rectangle. And, you know that the opposite sides of rectangles are congruent, or equal in length. So, if George Street is 100 meters, then so is Ringo Street. If John Street is 50 meters, so is Paul Street. Therefore, the perimeter of this quadrilateral, and the distance you ran, is 100 + 100 + 50 + 50, or 300 meters. That's a small block, less than one lap around the track.
If you know the length of all four sides of a quadrilateral, you can always find the perimeter by just adding them. If a side or two is missing, see if you can figure out the missing information using your knowledge of the shape's properties. Remember, with squares and rhombuses, all four sides are equal, so you only need to have one side given. With squares and parallelograms, the opposite sides are equal.
But, what happens when you get so good at running that you're ready to train for a marathon? That's 26.2 miles. You'll need to run a much bigger loop while you train, and it may not be such a tidy rectangle. It might look more like this:
Example of an irregular shape
If you need to know the perimeter of an irregular shape like this, then you really do need to know the lengths of each side.
Fortunately, you spotted helpful signs along your way, and you noted how many miles each street was. To find the perimeter, just add up each length. In miles, this example is 6 + 2 + 3 + 4 + 4 + 4 + 3, or 26 miles. That's almost a full marathon! If you add in that .2 mile-long detour you took to flag down an ice cream truck at the end, you just ran a marathon distance!
The next time you go for a long training run, the route doesn't have helpful signs to tell you how long each street is. If that's the case, you may be able to figure out the perimeter by treating your route as a group of combined shapes. Let's look at this example:
Example of combined shapes
We know the lengths of some of the sides, but not all of them. But, look closer at the shape. It's actually a combined shape comprised of a group of squares and rectangles.
In the top square, we know two sides are 4 miles, so the other side must also be 4 miles. As for the bit below it, well, we know the entire length of that rectangle is 10 miles. If two sections are 3 and 4, then the missing part must be 3 miles.
And, on the part on the side, the small length is going to be equal to the opposite side of the small rectangle, so it's 1 mile. Finally, the remaining part is just 5 minus 2, or 3 miles. So, now we know all the sides! It's 4 + 4 + 4 + 3 + 1 + 2 + 1 + 3 + 10 + 5 + 3, or 40 miles. Holy cow! That's way farther than you planned. I think you earned some ice cream.
Now, let's say you wake up the next day and all you can do is hobble around your house, leaning on walls for support. But, you still want to know how far you're moving, since every step hurts. Since you live in a geometry lesson, your house has some challenging geometric qualities. You can do this! You know a few parts, which is all you need. Just break it up into shapes like this:
Another example of combining shapes to find perimeter
Let's start with the big one. One side is 8 feet. Then, you can see that the two 4-foot stretches add up to 8, so that's a square. That means all sides are 8 feet long. As for the rectangle, well, one side is 4, so the opposite side is 4. If the other side is 2 and 3, then the missing piece is also 2.
Finally, those pesky triangles. But wait, those are special triangles. First, they're right triangles. But, more than that, look at the two legs of the top one. They are 3 and 4 feet, which makes it a 3-4-5 triangle. So, the hypotenuse is 5 feet. If you forget what a 3-4-5 triangle is, you could also use the Pythagorean theorem, or a^2 + b^2 = c^2. 3^2 + 4^2 is 25. The square root of 25 is 5.
Then, there's the final triangle. This is also a 3-4-5 triangle, so the missing side is 4 feet. If you put everything together, it's 8 + 8 + 8 + 5 + 2 + 4 + 2 + 5 + 4, or 46 feet. Between all that running, walking and geometry, I think you earned extra ice cream today.
In summary, if you know the lengths of all the sides of a shape, simply adding them together will give you the perimeter.
With a regular quadrilateral, like a square or a parallelogram, you just need to know one or two sides, and you can figure out the rest.
With an irregular shape, the perimeter is still just the sum of all the sides. If you don't know all the lengths, look to see if you can break it up into a group of combined shapes. Then use the properties of those shapes to determine the missing sides.
Following your completion of this lesson, you might be able to:
- Find the perimeter of regular quadrilaterals and irregular shapes
- Explain how to find the missing lengths of irregular shapes by combining shapes and using the properties of those combined shapes