# Trapezoid Perimeter Calculator

Created by Davide Borchia
Reviewed by Komal Rafay
Last updated: Mar 14, 2022

A trapezoid is a quadrangular shape, part of a broad family of convex quadrilaterals: use our trapezoid perimeter calculator to find its perimeter using any possible combinations of parameters!

## What is a trapezoid?

A trapezoid is a quadrilateral convex shape:

• Quadrilateral means that it has four sides (and four angles); and

• Convex means that it contains every segment with initial and final points inside the shape.

Trapezoids have at least a pair of parallel sides. The other sides can be either perpendicular or tilted with respect to the bases.

## The elements of a trapezoid

A characteristic set of elements identifies a trapezoid. Starting from the sides, we have:

• Two bases: the parallel sides;
• Two legs: the oblique sides;

It is possible to identify four angles. The total sum is always $360\ \degree$, which is equally divided between angles associated to the same side (complementary angles), which sum to $180\ \degree$.

⚠️ Don't mistake a trapezoid for a trapezium: the latter one is a quadrilateral without parallel sides!

It is important to identify one last element, the height of the trapezoid, the distance between the bases. It will be helpful in the calculations for the perimeter.

## How to calculate the perimeter of a trapezoid

The formula of a trapezoid's perimeter is extremely simple:

$P=a+b+c+d$

Where:

• $P$ is the perimeter;
• $a$ and $b$ the parallel sides (bases); and
• $c$ and $d$ the oblique sides.

But rarely, you will have all the sides. You may have combinations of sides and angles, and sometimes the situation may also look pretty dire until trigonometry comes to our help. Let's check the possible combinations you may find!

#### Two sides, an angle, and a base

With this combination you can find easily the height $h$ of the trapezoid with the formulas:

$h=c\cdot\sin{\alpha}$

or

$h=d\cdot\sin{\delta}$

Once you know the value of $h$, you can compute the value of the other angle, and then the projection of the sides on the base $a$, which summed equal the difference of the bases:

$a-b=c\cdot\cos{\alpha}+d\cdot\cos{\delta}$

Rearranging this formula gives you the missing base!

#### Two sides, a base, and the height

With this combination, we can first find the values of the projections of the sides on the base $a$ using the reliable Pythagorean theorem. Once you find them, you can either:

• Add them to the base $b$ to find the base $a$;
• Subtract them from the base $a$ to find the base $b$.

You have all the elements to feed into the trapezoid perimeter formula now!

#### Two angles associated to opposite sides, the height and a base

This case is similar to the previous one; knowing the height and the angles (remember that they satisfy the identities $\alpha+\beta=180\ \degree$ and $\gamma+\delta=180\ \degree$), you can find the sides first and then the projections on the base $a$. From there, you can find the value of the other base.

#### The two bases, a side, and the relative angle

This combination is our favorite! By subtracting the two bases and the projection of the side on $a$, we can find the other projection. From here, with the height calculated from the side and angle, we can find the other side.

⚠️ Using this combination, you will see the perimeter field in our calculator greyed out. You can't use our trapezoid perimeter calculator in reverse because of a slightly more complex formula running in the background. 🙄 You can try it in reverse in the other cases, however!

## How to use our trapezoid perimeter calculator

Using our trapezoid perimeter calculator is straightforward! If you have the values of each side of the polygon, insert them in the respective field, and you'll get the value of the perimeter.

🙋 You can use our calculators in reverse too! If you have the perimeter but one of the sides is missing, insert the value anyway, we will give you the answer. 😀

If you don't have the sides but some other combination of sides, angles, or height, click on advanced mode, you will then see all of the other variables. Choose the ones you need and insert your values there!

## Other trapezoid calculators you may find useful

Apart from the trapezoid perimeter calculator, we have many more calculators that will help you with your geometrical problems (only those, though!): try our:

## FAQ

### How to calculate the perimeter of a trapezoid?

The perimeter of a trapezoid formula is:
P = a + b + c + d
where P is the perimeter, a and b are the bases, while c and d are the sides. There are many ways to find the values of those elements: discover them on omnicalculator.com!

### What is the perimeter of a trapezoid with sides c=4 and d=3, base a=10 and α=30°?

First, you have to calculate the height of the trapezoid, using:
h = c * sin(α) = 4 * sin(30°) = 2.

Now using the Pythagorean theorem we can find the values of the projections of the sides on the base a:
sqrt(c² - h²) = sqrt(16 - 4) = 3.46 and sqrt(d² - h²) = 2.24.

Subtract these values from the base a to find b:
b = a - sqrt(c² - h²) - sqrt(d² - h²) = 10 - 3.46 - 2.24 = 4.3
and then the perimeter:
P = 10 + 4 + 3 + 4.3 = 21.3

### Can you find the perimeter of a trapezoid knowing only the angles and a base?

No. This combination of parameters would leave too much freedom to the remaining elements of the trapezoid: the other base's length would be able to vary as well as the angles of the trapezoid. It is necessary to know at least an angle in addition to these elements.

### How do I find the perimeter of an isosceles trapezoid?

In the case of an isosceles trapezoid, you can find the perimeter just by knowing the bases and one side: the two sides are identical! The formula changes slightly:
P = a + b + 2c
where P is the perimeter, a and b the bases, and c the oblique side

Davide Borchia
a (base)
in
b (base)
in
c (side)
in
d (side)
in
Perimeter
Perimeter (P)
in
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