how to find the perimeter of a trapezoid

A trapezoid i defined a a rectangular hape that ha two parallel ide. A with any polygon, to find the perimeter of the trapezoid, you mut add all of it four ide. However, often time you don't know

How to Find the Perimeter of a Trapezoid

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A trapezoid is defined as a rectangular shape that has two parallel sides. As with any polygon, to find the perimeter of the trapezoid, you must add all of its four sides. However, often times you don't know one side length, but have other information, such as the height of the trapezoid or the measurement of the angle. With this information, you can use the rules of geometry and trigonometry to find the length of the missing side.

Step

Method 1 of 3: If you know the lengths of both sides, top, and bottom

Set up the formula for perimeter of a trapezoid. The formula is, where is perimeter of the trapezoid, variable is the length of the top of the trapezoid, is the length of the bottom of the trapezoid, is the length of the left side of the trapezoid, and is the length of the right side of the trapezoid.
Plug the side lengths into the formula. If you don't know the lengths of the four sides of the trapezoid, you can't use this formula.
- For example, if you have a trapezoid with a top 2 cm long, a bottom 3 cm, and two sides 1 cm long, your formula will look like this:
Add up the length of the sides. This addition will give the perimeter of your trapezoid.
- For example:
  So, the perimeter of the trapezoid is 7 cm.

Method 2 of 3: Knowing the Height, Length of Both Sides, and Length of the Top

Divide the trapezoid into one rectangle and two right triangles. To do this, draw a high line from the tops of the two trapezoidal tops.
- If you can't form two right triangles because one side of the trapezoid is perpendicular to the base, note that this side is the same measurement as the height, and divide the trapezoid into one rectangle and one right triangle.
Write down the measurements for all the heights. Since these two lengths are opposite sides of a rectangle, they will be the same length.
- For example, if you have a trapezoid that is 6 cm high, you will need to draw a line from each top to the bottom of the trapezoid. Write the measurement of the line as 6 cm.
Write down the measurement of the length from the center on the base of the trapezoid. (This is the base of the rectangle). It is the same length as the top (top of the rectangle) because the opposite sides of the rectangle are the same length. If you don't know the length of the top, you can't use this method.
- For example, if the top of the trapezoid is 6 cm long, the length of the center at the base of the trapezoid is also 6 cm.
Set up the Pythagorean Theorem formula for the first right triangle. The formula is, where is the length of the hypotenuse or hypotenuse of a right triangle (the side opposite the right angle), is the height of the right triangle, and is the length of the base of the triangle.
Plug in the known values for the first triangle into the formula. Make sure you plug in the side length of the trapezoid for the variable. Enter the height of the trapezoid for the variable.
- For example, if you know that the height of the trapezoid is 6 cm and the length of the hypotenuse of the trapezoid is 9 cm, your equation will look like this:
Square the known values in the equation. Then, subtract to isolate the variable.
- For example, if the equation is, you would square 6 and 9, then subtract 6 from the square 9:
Take the square root to find the value math> b. (For complete instructions on how to simplify a square root, you can read Simplify a Square Root.) This calculation will give you the unknown length of the base of your first right triangle. Write down the measurements for the base of your triangle.
- For example::
  So, you will write on the base of your first triangle.
Find the unknown length of the second right triangle. To do this, write down the Pythagorean Theorem formula for the second triangle, and follow the steps to find the length of the missing side. If you are working with an isosceles trapezoid, which is a trapezoid where the two non-parallel sides are the same length, the two right triangles are congruent, so you can write down the values of the first triangle equal to the values of the second triangle.
- For example, if the second side of the trapezoid is 7 cm, you would calculate:
  So you must write on the base of your second triangle.
Add up all the side lengths of the trapezoid. The perimeter of any polygon is the sum of all sides :. For the bottom base, you will add the bottom rectangle and the bases of the two triangles. Chances are, you will have the square root in your answer. For complete instructions on how to add square roots, you can read the article on adding square roots. You can also use a calculator to convert square roots to decimals.
- For example,
  Converting the square root to a decimal, you have
  So, your approximate perimeter of the trapezoid is 38.314 cm.

Method 3 of 3: If you know the height, top length, and bottom inner angle

Divide the trapezoid into one rectangle and two right triangles. To do this, draw a high line from the top of the trapezoid.
- If you can't form two right triangles because one side of the trapezoid is perpendicular to the base, note that this side is the same measurement as the height, and divide the trapezoid into one rectangle and one right triangle.
Write down the measurements for all the heights. Since these two lengths are opposite sides of a rectangle, they will be the same length.
- For example, if you have a trapezoid that is 6 cm high, you will need to draw a line from the top to the bottom of the trapezoid. Write down the measurement of the line as 6 cm.
Write down the measurement of the length from the center on the base of the trapezoid. (This is the base of the rectangle). It is the same length as the top (top of the rectangle) because the opposite sides of the rectangle are the same length.
- For example, if the top of the trapezoid is 6 cm long, the length of the center at the base of the trapezoid is also 6 cm.
Write down the sine ratio for the first right triangle. The ratio is, where is the measurement of the inner angle, is the height of the triangle, and is the length of the hypotenuse or slash.
- This ratio will allow you to find the length of the hypotenuse of the triangle, which is also the length of the first side of the trapezoid.
- The hypotenuse is the side opposite the 90 degree angle of a right triangle.
Plug in the known values into the sine ratio. Make sure you use the height of the triangle for the "front" side in the formula. You will be looking for an H.
- For example, if you know that the inner angle is 35 degrees and the height of the triangle is 6 cm, your formula will look like this:
Find the sine of the angle. Do this by using the SIN button on a scientific calculator. Plug this value into the comparison.
- For example, on a calculator, you will find that the sine of the 35 degree angle is 0.5738 (rounded). So, your formula will now be:
Find an H. To do this, multiply each side by H, then divide each side by the sine of the angle. Or, you can divide the height of the triangle by the sine of the angle.
- For example:
  So, the length of the hypotenuse and the first unknown side of the trapezoid is about 10.4566 cm.
Find the length of the hypotenuse for the second right triangle. Write down the sine ratio () for the second inner angle. This calculation will give you the length of the hypotenuse, which is also the side of the second trapezoid.
- For example, if you found that the inner angle was 45 degrees, you would calculate:
  So, the length of the hypotenuse and side of the second unknown side of the trapezoid is about 8.4854 cm.
Set up the Pythagorean Theorem formula for the first right triangle. The formula for the Pythagorean Theorem is, where the length of the hypotenuse is and the height of the triangle is.
Plug the values you know into the Pythagorean Theorem for the first right triangle. Make sure you plug in the length of the hypotenuse for and the height of the triangle for.
- For example, if the first right triangle has a hypotenuse of 10.4566 and a height of 6, your formula will be:
Look for value. This calculation will give you the base length of the first right triangle as well as the unknown portion of the bottom base of the first trapezoid.
- For example:
  So, the base of the triangle and the bottom base of the first unknown trapezoid is about 8.5639 cm.
Find the unknown length of the base of the second right triangle. Use the Pythagorean Theorem () formula to find it. Plug in the length of the hypotenuse for and the height for the triangle to. Finding the value will give you the unknown length of the bottom base of the trapezoid.
- For example, if the second right triangle has a hypotenuse of 8.4854 and a height of 6, you would calculate:
  So, the base of the second triangle and the missing bottom base of the trapezoid is 6 cm.
Add up all the side lengths of the trapezoid. The perimeter of any polygon is the sum of all its sides:. For the bottom base, you will add the bottom sides of the rectangle and the bases of the two triangles.
- For example,
  So, your approximate perimeter of the trapezoid is 45.5059 cm.

Tips

Use the custom triangle rule to find the unknown length of a special triangle without using sine or the Pythagorean Theorem. This rule applies to triangles 30-60-90, or triangles 90-45-45.
Use a scientific calculator to find the sine of an angle by entering the angle measurement and pressing the "SIN" button. You can also use trigonometric tables.

Things You Will Need

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