How to Find the Missing Side of a Trapezoid

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A trapezoid is defined as a quadrilateral with two parallel sides. As with any polygon, to find the perimeter of a trapezoid you need to add all four of its sides together. However, often you will be missing side lengths but have other information, such as the height of the trapezoid, or the angle measurements. Using this information, you can use rules of geometry and trigonometry to find the unknown lengths of sides.

1. 1

   Set up the formula for perimeter of a trapezoid. The formula is ${\displaystyle P=T+B+L+R}$ , where ${\displaystyle P}$ equals the perimeter of the trapezoid, and the variables ${\displaystyle T}$ equals the length of the top base of the trapezoid, ${\displaystyle B}$ equals the length of the bottom base, ${\displaystyle L}$ equals the length of the left side, and ${\displaystyle R}$ equals the length of the right side.^[1]

2. 2

   Plug the side lengths into the formula. If you do not know the length of all four sides of the trapezoid, you cannot use this formula.

   • For example, if you have a trapezoid with a top base of 2 cm, a bottom base of 3 cm, and two side lengths of 1 cm, your formula will look like this:
     ${\displaystyle P=2+3+1+1}$

3. 3

   Add the side lengths together. This will give you the perimeter of your trapezoid.

1. 1

   Divide the trapezoid into a rectangle and two right triangles. To do this, draw the height from both top vertices.

   • If you cannot form two right triangles because one side of the trapezoid is perpendicular to the base, just note that this side will have the same measurement as the height, and divide the trapezoid into one rectangle and one right triangle.

2. 2

   Label each height line. Since these are opposite sides of a rectangle, they will be the same length.^[2]

   • For example, if you have a trapezoid with a height of 6 cm, you should draw a line from each top vertex extending down to the bottom base. Label each line 6 cm.

3. 3

   Label the length of the middle section of the bottom base. (This is the bottom side of the rectangle.) The length will equal the length of the top base (the top side of the rectangle), because opposite sides of a rectangle are of equal length.^[3] If you do not know the length of the top base, you cannot use this method.

   • For example, if the top base of the trapezoid is 6 cm, then the middle section of the bottom base is also 6 cm.

4. 4

   Set up the Pythagorean Theorem formula for the first right triangle. The formula is ${\displaystyle a^{2}+b^{2}=c^{2}}$ , where ${\displaystyle c}$ is the length of the hypotenuse of the right triangle (the side opposite the right angle), ${\displaystyle a}$ is the height of the right triangle, and ${\displaystyle b}$ is the length of the base of the triangle.^[4]

5. 5

6. 6

   Square the known values in the equation. Then, subtract to isolate the ${\displaystyle b}$ variable.

7. 7

   Take the square root to find the value of $$ b {\displaystyle b} . (For complete instructions on how to simplify square roots, you can read Simplify a Square Root.) The result will give you the value of the missing base of your first right triangle. Label this length on the base of your triangle.

8. 8

   Find the missing length of the second right triangle. To do this, set up 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 in which the two non-parallel sides are the same length,^[5] the two right triangles are congruent, so you can simply carry the value from the first triangle over to the second triangle.

9. 9

   Add up all the side lengths of the trapezoid. The perimeter of any polygon is the sum of all sides: ${\displaystyle P=T+B+L+R}$ . For the bottom base, you will add the bottom side of the rectangle, plus the bases of the two triangles. You will likely have square roots in your answer. For complete instructions on how to add square roots, you can read the article Add Square Roots. You can also use a calculator to convert the square roots to decimals.

1. 1

   Divide the trapezoid into a rectangle and two right triangles. To do this, draw the height from both top vertices.

   • If you cannot form two right triangles because one side of the trapezoid is perpendicular to the base, just note that this side will have the same measurement as the height, and divide the trapezoid into one rectangle and one right triangle.

2. 2

   Label each height line. Since these are opposite sides of a rectangle, they will be the same length.^[6]

   • For example, if you have a trapezoid with a height of 6 cm, you should draw a line from each top vertex extending down to the bottom base. Label each line 6 cm.

3. 3

   Label the length of the middle section of the bottom base. (This is the bottom side of the rectangle.) This length will be equal to the length of the top base, because opposite sides of a rectangle are of equal length.^[7]

   • For example, if the top base of the trapezoid is 6 cm, then the middle section of the bottom base is also 6 cm.

4. 4

5. 5

   Plug the known values into the sine ratio. Make sure you use the height of the triangle as the length of the opposite side in the formula. You will solve for H.

   • For example, if the given interior angle is 35 degrees, and the height of the triangle is 6 cm, your formula will look like this:
     ${\displaystyle \sin(35)={\frac {6}{H}}}$

6. 6

   Find the sine of the angle. Do this by using the SIN button on a scientific calculator. Plug this value into the ratio.

   • For example, by using a calculator you will find that the sine of a 35 degree angle is .5738 (rounded). So your formula will now be:
     ${\displaystyle .5738={\frac {6}{H}}}$

7. 7

   Solve for H. To do this, multiply each side by H, then divide each side by the angle sine. Or, you can simply divide the height of the triangle by the angle sine.

8. 8

   Find the length of the hypotenuse of the second right triangle. Set up the sine ratio ( ${\displaystyle \sin \theta ={\frac {\text{opposite}}{\text{hypotenuse}}}}$ ) for the second given interior angle. This will give you the length of the hypotenuse, which is also the first side of the trapezoid.

9. 9

   Set up the Pythagorean Theorem formula for the first right triangle. The Pythagorean Theorem formula is ${\displaystyle a^{2}+b^{2}=c^{2}}$ , where the length of the hypotenuse is ${\displaystyle c}$ , and the height of the triangle is ${\displaystyle a}$ .

10. 10

11. 11

    Solve for $$ b {\displaystyle b} . This will give you the length of base of the first right triangle, and the first missing section of the trapezoid's bottom base.

12. 12

13. 13

    Add up all the side lengths of the trapezoid. The perimeter of any polygon is the sum of all sides: ${\displaystyle P=T+B+L+R}$ . For the bottom base, you will add the bottom side of the rectangle, plus the bases of the two triangles.

    • For example, ${\displaystyle 6+(8.5639+6+6)+10.4566+8.4854=45.5059}$
      So, the approximate perimeter of your trapezoid is 45.5059 cm.

Add New Question

• Question

  How can I solve the hypotenuse of a right triangle with a height of 2ft?

  

  You don't have enough information to find the hypotenuse. You would need the lengths of both legs or the size of at least one of the acute angles or the area of the triangle.

• Question

  How do I find the area without knowing the length of the sides of the trapezoid?

  

  You would have to know the height of the trapezoid (h) and the lengths of both parallel sides (a and b). The area formula is [h(a + b)] / 2.

• Question

  Why are there so many formulas?

  

  It's because there are several possible sets of known dimensions regarding a trapezoid.

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• Use the laws of special triangles to find the missing lengths of special triangles without using sine or the Pythagorean Theorem. The laws apply to a 30-60-90 triangle, or a 90-45-45 triangle.

• Use a scientific calculator to find the sine of an angle by entering the angle measurement, then hitting the "SIN" button. You can also use a trigonometry table.

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Article SummaryX

To find the perimeter of a trapezoid if you know the length of both sides and the bases, add together the length of all 4 sides. If you know the height, both side lengths, and the top base length, draw a straight line down from each top corner to form a square and 2 triangles. Then, use the Pythagorean Theorem to find the length of the base of each triangle. Add the length of each triangle base to the length of the top base, then add that to the top base and both sides to get the perimeter. To learn more about using the Pythagorean Theorem, keep reading!

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How to Find the Missing Side of a Trapezoid

Source: https://www.wikihow.com/Find-the-Perimeter-of-a-Trapezoid

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