A trapezoid, also called a trapezium in some countries, is a geometric shape with one pair of parallel sides. It looks like a complex shape when you’re trying to figure out things like area, perimeter, and angles, but it is actually quite simple when you break it down into its basic part, then use a few mathematical formulas to derive other quantities.
Perimeter, as you may recall from geometry, is nothing but the sum of the length of all the sides of a shape.
There are some instances in geometry, algebra, calculus, and practical applications that will require you to find the perimeter of a trapezoid when you may not necessarily have all the quantities. When doing a trapezoid calc: find P operation, you can add up the known sides and figure out the rest.
Properties of Trapezoids
The only constant property of trapezoids is the existence of a single pair of parallel sides. Beyond that, a trapezoid can take any configuration. The two parallel sides can have any difference in length, or they can be the same length. In other words, a rectangle or a square is technically a trapezoid, just a special type of one.
Finding the Perimeter of a Simple Trapezoid
To find the perimeter of a trapezoid (or any shape except an ellipse or a circle) you just have to add the lengths of all the sides. However, if you don’t know the side lengths, you might have a problem. By using properties of geometry, trigonometric identities, and basic logic, you can infer or find the values of the unknown sides of a trapezoid.
For example, if you know the angle formed by the base and one of the sides, and if that angle is the same on the other side, you can then infer that the opposite side must be the same length because it has to cover the same distance to reach the parallel top of the trapezoid. However, you may not be given this information outright and you’ll have to figure it out on your own.
The best way to think of a trapezoid is as a rectangle with two right triangles on the sides. Then, by using the properties of right triangles, you can figure out the length of the unknown side.
We’ll start by working with a trapezoid ABCD, where each letter represents a vertex where two line segments intersect. We’ll assume, that line segment AB and line segment CD are parallel. AB has a length of 5, and CD has a length of 11. We also know that the trapezoid has a height of 4, if you were to take a perpendicular line connecting AB to CD.
Finally, angles ACD and BCD are given to be the same.
Because we have the top base has a length of 5, you can think of the bottom base as including a rectangle with a length of 5. Because we know that the angle is the same for both sides, you know that the two sides have to be the same length. Therefore, you now have to imagine the trapezoid broken down into other shapes.
The height is 4, and the total length of the bottom base CD is 11. Because AC and BD are the same lengths, it makes things easier. Now, on each side of a length of 5 on CD, you have 6 unaccounted for, so you can think of a right triangle with 3 on one leg and 4 on the other because you already know that height.
The next step in trapezoid calc: find p is to work out the length of the right triangle’s hypotenuse. The formula for the Pythagorean theorem is a2 + b2 = c2. Square the lengths of the two legs.
The square of 3 is 9, and the square of 4 is 16. Adding these up, you get 25. Now, take the square root, which is 5. In this example, we’ve used the simplest right triangle there is.
It won’t always be this simple, but it serves as an example. Because you know that the two right triangles are the same dimensions, you can infer that the hypotenuse is the same on both sides.
Finally, you can go back to your original trapezoid and substitute the two values into the original shape. Then all you have to do is add the sides as you normally would.
In this example, 5 + 5 + 5 + 11, for a total of 26. Therefore, the trapezoid in our example has a perimeter of 26.
Using Trigonometric Identities
There may be times when you have to use some knowledge of trigonometry and a scientific calculator to help you out.
Here are the three identities you’ll use most often:
The tangent is also equal to the sine divided by the cosine.
You can also use the Law of Sines: a/sin(A) = b/sin(B) = c/sin(C).
This is how you’ll find any unknown angles or, with the help of some algebra, the sides if you need them.
We’ll assume for now that you’re just working with a right triangle with another known angle of 60 degrees and a hypotenuse 5 units long, and you want to find the lengths of the other sides. Remember that as per geometry, the three angles of any triangle add up to 180 degrees. Because one of them is by default 90 degrees, you have 90 degrees to account for.
You know your angles: a/sin(60) = b/sin(30) = 5/sin(90). Simplifying, you get b/0.5 = 5. Therefore, b equals 10. You can get the other sides this way as well.
Finding Perimeter of a Trapezoid on a Graph
Having the figures sketched out in geometric diagrams is one thing, but having it on a graph coordinate system can make it even easier to find the perimeter. Assuming that each increment of x and y is one unit, the only part for which you might have a problem is the hypotenuse. However, even that is easy to calculate.
All you have to do is count the units of x and y that make up the right triangle, then square the units and take the square root of the result. This gives you the diagonal of the trapezoid. To get the lengths of the sides, all you have to do is count the units down to the x-axis and across to the other parallel line.
What if You Don’t Have a Graph?
Even if you don’t have a graph and are just given points, you can do simple subtraction. For example, take the points (0, 3) and (0, 9). To determine the length of the line segment and the slope, you use the formula (y1 – y0)/(x1 – x0). Substituting the values, you get 0 and 6.
The formula above is meant for slope, but it’s also an easy way to visualize. If your x-values are the same, you have a vertical line.
Now, you need to find your diagonal line of the trapezoid. It doesn’t matter what slope it is; all you’re looking for is the length. By finding the change in y and the change in x, you get two sides of your right triangle. You should have a good idea of what to do from here: square the lengths of the two sides, add them, and take the square root of the result.
Now, unless you were fortunate enough to get a rational root – i.e., one that simplifies to a fraction or decimal, you’re not going to be able to write it as a standard integer. For example, say that you got a diagonal with a length √2. You would have to add the other sides on their own, such as 3 + 3 + 5, which gives you 11. Then, you add √2.
From here, there’s nothing further you can do. If you have something like √8, you can simplify that into 2√2, which is √2 * √4. Always reduce math expressions to their simplest forms when you can. If you’re in school, odds are you’ll get only partial credit if you don’t simplify your answers.
Learning to solve trapezoid calc: find P problems is far easier if you learn first to do it algebraically instead of relying on a graph. Not all graphs will resolve to neat points and therefore risk giving you inaccurate answers.
The perimeter of a trapezoid is one step in helping you to find its area. Because the area is a function of height and width, you need to gather as many concrete measurements as you can. If you’re working on a graph, it might help to visualize the trapezoid as lying sideways, with the vertical lines being the bases and the horizontal lines being the height.
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