Welcome to Omni's trapezoid calculator, where we'll learn all about these four-sided shapes. We'll show you how to calculate the area of a trapezoid, how to find the height of a trapezoid, or what the trapezoid perimeter formula looks like. Also, we'll take the time to describe some special types of the quadrangle: the isosceles trapezoid and the right trapezoid. And don't worry; we leave no stone unturned - we even mention the median and the trapezoid angles in the calculator.
It looks like there are quite a few things to discuss, so let's get going, shall we?
What is a trapezoid?
A trapezoid is a quadrangle (a shape that has four sides) which has at least one pair of opposite sides parallel to each other. Note that we said "at least one pair of sides" - if the shape has two such pairs, it's merely a rectangle. And make no mistake - every rectangle is a trapezoid. The inverse is, of course, not true.
The two sides, which are parallel, are usually called bases. Usually, we draw trapezoids the way we did above, which might suggest why we often differentiate between the two by saying bottom and top base. The two other non-parallel sides are called legs (similarly to the two sides of a right triangle).
There are a few special cases of trapezoids that we'd like to mention here.
We've already mentioned that one at the beginning of this section - it is a trapezoid which has two pairs of opposite sides parallel to one another.
A trapezoid whose legs have the same length (similarly to how we define isosceles triangles).
A trapezoid whose one leg is perpendicular to the bases. Firstly, note how we require here only one of the legs to satisfy this condition - the other may, or may not. Secondly, observe that if a leg is perpendicular to one of the bases, then it is automatically perpendicular to the other as well since the two are parallel.
With these special cases in mind, a keen eye might observe that rectangles satisfy conditions 2 and 3. Indeed, if someone didn't know what a rectangle is, we could just say that it's an isosceles trapezoid which is also a right trapezoid. Quite a fancy definition compared to the usual one, but it sure makes us sound sophisticated, don't you think?
Before we move on to the next section, let us mention two more line segments that all trapezoids have.
The height of a trapezoid is the distance between the bases, i.e., the length of a line connecting the two, which is perpendicular to both. In fact, this value is crucial when we discuss how to calculate the area of a trapezoid, and therefore gets its own dedicated section.
The median of a trapezoid is the line connecting the midpoints of the legs. In other words, with the above picture in mind, it's the line cutting the trapezoid horizontally in half. It is always parallel to the bases and with notation as in the figure, we have
median = (a + b) / 2. If you're curious about the name, make sure to check out Omni's median calculator (note: it doesn't concern trapezoids).
Alright, we've come to know our shape quite well; we even saw one trapezoid formula! Let's go one step further and try to understand the topic even better. We begin this in-depth analysis with the trapezoid perimeter formula and its inside angles.
Trapezoid perimeter formula and the trapezoid angles
The perimeter of a polygon is the sum of its side lengths. For the hero of today's article, the story is no different. With notation as in the picture in the first section (and in the trapezoid calculator), we deduce the trapezoid perimeter formula to be:
P = a + b + c + d.
Pretty simple, wouldn't you say?
Next, let's talk about angles. Just like any other quadrangle, the sum of angles in a trapezoid is
360 degrees (or
2π radians). However, the condition for being a trapezoid (i.e., having a pair of parallel sides) forces additional properties on the individual ones. To be precise, the couple of angles along one of the legs are supplementary angles. This means that their sum must equal
180 degrees (or
π radians), which in notation from the figure in the first section, translates to:
α + 𝛾 = β + δ = 180°.
Note how our tool also mentions angles in the bottom set of variable fields. As such, it can also serve as a trapezoid angle calculator for whenever these are the numbers we're seeking. And indeed, they often come in handy - they play an essential role when we learn how to find the height of a trapezoid, and that, in turn, appears when studying how to calculate the area of a trapezoid. However, let us begin with the latter question.
How to calculate the area of a trapezoid
Let us have the picture from the first section again so that you don't have to scroll through the whole article whenever you'd like to recall notation.
The area of a trapezoid formula is as follows:
A = (a + b) * h/2.
Note that indeed, just as we mentioned a couple of times already, it's crucial to know how to find the height of a trapezoid to compute its area. Also, the legs never appear in the equation. Sure, they define the shape of our quadrangle, but their lengths are used only in the trapezoid perimeter formula we've discussed in the above section.
Lastly, let us make it clear that by order of operations, it doesn't matter at what moment we divide by
2 in the above area of a trapezoid formula. We can either calculate
(a + b) * h first and then divide the whole thing by
2, or first find
h/2, and only then multiply it by
(a + b). In fact, a keen eye will observe that
(a + b) / 2 is the median we've mentioned in the first section. In other words, alternatively, we can use the formula
A = median * h to find
Alright, we've learned how to calculate the area of a trapezoid, and it all seems simple if we they give us all the data on a plate. But what if they don't? The bases are reasonably straightforward, but what about
h? Well, it's time to see how to find the height of a trapezoid.
How to find the height of a trapezoid
The crucial fact we use to find the height of a trapezoid is that it is a line segment perpendicular to the bases. That gives us a right angle at both endpoints, which allows us to use right triangles. And the first thing that comes to mind when we hear the phrase right triangle is, of course, the Pythagorean theorem.
Let's draw a line from one of the top vertices that falls on the bottom base
a at an angle of
90 degrees. (Observe how for obtuse trapezoids like the one in the right picture above the height
h falls outside of the shape, i.e., on the line containing
a rather than
a itself. Nevertheless, what we describe further down still holds for such quadrangles.) The length of this line is equal to the height of our trapezoid, so exactly what we seek. Note that by the way we drew the line, it forms a right triangle with one of the legs
d (depending which top vertex we chose).
If we have the length of a trapezoid's leg and can figure out the other side of the right triangle (i.e.,
f in the picture above), then we know how to find the height of a trapezoid - we use the Pythagorean theorem. However, there is also one other way to calculate it.
If you know a little about trigonometry, you'll be able to find the height using the inside angle of the trapezoid. To be precise, looking at the trapezoid angles in our calculator (i.e., at the notation in the picture), we can use the definition of trigonometric functions to write:
h = c * sin(α) = d * sin(δ),
sin is the sine function. As a matter of fact, it may happen that the angle is equal to
60 degrees, in which case we can simply use the properties of the special right triangles with such inside angles.
Lastly, let us mention that all this search for
h is super easy in a special case - when we have a right trapezoid. Then the height of our trapezoid is simply the leg that lies next to the right angle. Note that in this case the above trigonometric formula still works since
sin(90°) = 1.
Phew, that was a lot of theory. It's high time we use these trapezoid formulas and see how to calculate the area and perimeter of a trapezoid in practice.
Example: using the trapezoid calculator
Let's check out how to find the area and perimeter of a trapezoid with sides and angles denoted like in the trapezoid calculator, and the following data:
a = 8 in,
b = 5 in,
d = 3 in,
α = 90°,
δ = 45°.
It doesn't look like much, but let's see what we can do here. Firstly, however, let's observe that our trapezoid calculator can easily handle our problem even with so little information. Indeed, if we input the above numbers into our tool (note how we can switch to other units by clicking on them and choosing the suitable one from the list), it will fill out all other fields. For instance, as a trapezoid angle calculator, it will use the identities mentioned in the second section to compute
𝛾. Also, note that we can additionally go into the advanced mode and see the length of the median.
If the tool can do it, so can we! Let's see how to calculate the area and perimeter of a trapezoid by hand.
First of all, observe that we're dealing with a right trapezoid since
α = 90° (in fact, we also have
β = 90°). That means that side
c is perpendicular to the bases, and therefore equal to the height
c = h. However, we don't know
c, so we'll have to find it still.
To do this, draw the height of our trapezoid that goes from the vertex between
d. Together with
d and a part of
a, it forms a right triangle. What is more, we know one of its angles -
δ = 45°. That means that it is one of the special cases - it's half of a square. Therefore,
h is equal to the bottom side of the triangle, and
d is, in fact, the diagonal of a square, which means that:
h = d / √2 = 3 in / √2 = 1.5√2 in ≈ 2.1213 in
(we get the last equality by rationalizing the denominator).
Now we have everything needed to find
A. Recall from the dedicated section how to calculate the area of a trapezoid and use the information to obtain
A = (a + b) * h/2 = (8 in + 5 in) * 1.5√2 in / 2 = 9.75√2 in² ≈ 13.789 in².
We've also gathered all the data to find
c = h = 1.5√2 in. By the trapezoid perimeter formula from the second section we get
P = a + b + c + d = 8 in + 5 in + 1.5√2 in + 3 in = 16 + 1.5√2 in ≈ 18.12 in.
Not too bad, was it? The sides and angles we got at the beginning seemed like a pretty random bunch, but we've managed to put them to good use. If you feel hungry for more geometry and formulas, make sure to check out other 2D shape calculators at Omni's website - we have them all!