3 ways to calculate the perimeter of a trapezoid

3 ways to calculate the perimeter of a trapezoid
3 ways to calculate the perimeter of a trapezoid
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In geometry, the perimeter of a trapezoid is the sum of the length of its sides. When all the information is available, this calculation is easy. Otherwise, the calculation of the perimeter requires several intermediate steps in order to find the missing dimensions. The use of mathematical theorems or trigonometry may then be necessary. Regarding the notations used in the article, know that the length of a segment [AB] is written AB.

Steps

Method 1 of 3: Calculate the perimeter of a trapezoid knowing all its dimensions

Find the Perimeter of a Trapezoid Step 1

Step 1. Write down the formula for calculating the perimeter of an ABCD trapezoid

Let [DC] be the large base, [AB] the small base, [AD] and [BC] the other two sides of the polygon. As with all quadrilaterals, the perimeter P of the trapezoid is equal to the sum of the length of the four sides. In this case, it is given by the following formula: P = AB + BC + CD + DA.

Find the Perimeter of a Trapezoid Step 2

Step 2. Incorporate the numeric values ​​into the formula

To calculate the perimeter of a trapezoid in one step, you need to have its four lengths.

Assume the following lengths, expressed in centimeters: DC = 3, AB = 2, AC = BC = 1. In this particular case, note that it is an isosceles trapezoid, because its non-parallel sides have the same length

Find the Perimeter of a Trapezoid Step 3

Step 3. Add up the lengths

If we take the numerical values, the perimeter of the trapezoid is 7 cm.

The detailed calculation is P = 3 + 2 + 1 + 1 = 7

Method 2 of 3: Calculate the perimeter of a trapezoid using the Pythagorean theorem

Find the Perimeter of a Trapezoid Step 4

Step 1. Break down your trapezoid into three simpler figures

Let ABCD be a trapezoid of which you know the height, the dimensions of the small base [AB] as well as of the non-parallel sides [AD] and [BC]. The length of the large base [DC] is therefore the unknown that you need to find. To do this, you have to decompose the polygon. The trapezoid is a special quadrilateral because its bases are parallel. It is thus possible to divide it into three simple figures, namely a rectangle framed by two right triangles. To represent them, draw the two heights resulting from the vertices A and B.

To calculate the perimeter of a rectangular trapezoid, divide it into two simple shapes. Indeed, it has at least one right angle and can be broken down into a rectangle and a right triangle

Find the Perimeter of a Trapezoid Step 5

Step 2. Name the vertices of your figure

In addition to points A, B, C and D of the initial trapezoid, add points H1 and H2. They represent the intersections between the heights and the large base. You thus obtain the BCH triangles2 and ADH1.

  • The bases of the trapezoid being parallel, the two heights have the same length. This is also the width of the rectangle. Suppose the height of the trapezoid is 6 cm. So you have AH1 = BH2 = 6.
Find the Perimeter of a Trapezoid Step 6

Step 3. Note the dimensions of the ABH rectangle2H1.

The cutting of the trapezoid makes it possible to deduce them easily. The length of the rectangle is equal to that of the small base of the trapezoid, and its width is equal to the height.

  • Suppose the small base is 6 cm. The length of the rectangle is also 6 cm, which gives AB = H2H1 = 6. In this particular case, note that the polygon ABH2H1 is a square. That said, this observation does not impact the calculation.
Find the Perimeter of a Trapezoid Step 7

Step 4. Use the Pythagorean theorem to calculate the dimensions of the BCH triangles2 and ADH1.

In a right triangle, the hypotenuse is the side opposite the right angle, which is defined by a height and the associated base. The Pythagorean theorem states that, in the case of a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Note that, by abuse of language, the length of the side is assimilated to said side.

Find the Perimeter of a Trapezoid Step 8

Step 5. Apply the Pythagorean theorem to the triangle BCH2.

Let [BC] be its hypotenuse [BH2] its height and [CH2] its base. According to the Pythagorean theorem, the ratio of the lengths gives BC2 = BH22 + CH22.

  • Note that the hypotenuse of triangle BCH2 is also the [BC] side of the trapezoid. So you have the lengths of the hypotenuse and one of the sides of the triangle.
Find the Perimeter of a Trapezoid Step 9

Step 6. Isolate the unknown dimension of the triangle

To do this, refine the formula of the Pythagorean theorem. So you have CH22 = BC2 - BH22. So CH2 = √ (BC2 - BH22).

  • Suppose BH2 = 6 and BC = 9. By inserting these data in the previous formula, you get CH22 = (92 - 62) = 45.
Find the Perimeter of a Trapezoid Step 10

Step 7. Calculate the length of the base of triangle BCH2.

Take the formula established in the previous step. By integrating the digital data, you get CH22 = 45 or CH2 = √45. We must then simplify this result.

Other than perfect squares, the square root of a number is rarely an integer. For example, the square root of 45 is 6, 71. The simplified result is therefore √45 = √ (9 x 5) = √9 x √5 = 3√5

Find the Perimeter of a Trapezoid Step 11

Step 8. Calculate the length of the base of the triangle ADH1.

Let [AH1] its height [AD] its hypotenuse and [DH1] its base. To calculate the length of the base, apply the formula of the Pythagorean Theorem again. For the ADH triangle1, you have AD2 = AH12 + DH12. By isolating the base, you get DH12 = AD2 - AH12, which gives DH1 = √ (AD2 - AH12). Note that, if you are looking for the perimeter of an isosceles trapezoid, the side sides are the same length. By breaking down this figure, you get two identical triangles, which makes the calculations easier.

  • Suppose AH1 = 6 and AD = 7. This gives DH1 = √(72 - 62) = √13.
Find the Perimeter of a Trapezoid Step 12

Step 9. Calculate the perimeter of your trapezoid

Calculate the length of the large base. Based on the previous results, you have DC = DH1 + H1H2 + H2C. The formula for the perimeter P is therefore P = DH1 + H1H2 + H2C + AB + BC + AD. It's best to keep your values, especially square roots, as an integer.

By replacing with the numerical values, you get DC = √13 + 6 + 3√5 and P = √13 + 6 + 3√5 + 6 + 9 + 7 = 28 + 3√5 + √13. The perimeter of the trapezoid is therefore 38.3 cm (rounded to the nearest tenth)

Method 3 of 3: Calculate the perimeter of a trapezoid using trigonometry

Find the Perimeter of a Trapezoid Step 13

Step 1. Divide the trapezoid into three simple figures

Consider a trapezoid ABCD of which you know the height, the dimension of the small base [AB] and the value of the angles defined by the large base. As in the previous method, the decomposition of the trapezoid facilitates the calculation. Draw the heights from the vertices A and B of your trapezoid.

  • Note H1 and H2 the points of intersection between the heights and the large base. Your trapezoid is thus divided into a rectangle ABH2H1 and two right triangles, noted BCH2 and ADH1.
Find the Perimeter of a Trapezoid Step 14

Step 2. Deduce the width of the rectangle

Since the bases of the trapezoid are parallel, the two heights are the same length. It is also the width of the rectangle.

  • Suppose the height is 6cm. So you have AH1 = BH2 = 6.
Find the Perimeter of a Trapezoid Step 15

Step 3. Deduce the length of rectangle ABH2H1.

It is equal to that of the small base [AB].

  • Suppose the small base is 6 cm. So you have AB = H1H2 = 6. In this particular case, the width and length of the rectangle are the same, making it a square.
Find the Perimeter of a Trapezoid Step 16

Step 4. Calculate the dimensions of the non-parallel sides of the trapezoid

Given your data, you can find it through trigonometry. Indeed, the non-parallel sides of the trapezoid merge with the hypotenuse of the right triangles that compose it. Now, in a right triangle, the sine of an angle is equal to the ratio between the lengths of the side opposite the angle and of the hypotenuse.

  • You have the value of the angles in C and D as well as the height of the trapezoid. Now, in the BCH triangles2 and ADH1, the segments [BH2] and [AH1] are respectively the sides opposite to these angles. You can therefore calculate the length of their hypotenuse using the sine formula.
  • Trigonometry is an interesting tool in geometric calculations. Besides the sine, you can also use the cosine or tangent of an angle to find a dimension in a right triangle.
Find the Perimeter of a Trapezoid Step 17

Step 5. Calculate the length of the hypotenuse of triangle BCH2.

Let [BC] be the hypotenuse of triangle α2 the angle in C and [BH2] the side opposite this angle. By taking the trigonometric relation, you have sin α2 = BH2 / BC.

  • Suppose the angle in C is 35 °. This gives sin 35 = BH2 / BC.
Find the Perimeter of a Trapezoid Step 18

Step 6. Calculate the sine of the angle at C

There are only a few remarkable values ​​in trigonometry. For others, you have to use a scientific calculator.

In our example, the sine of the angle at C is 0.57. You can keep the integer value including the sine or round it off. The accuracy of the rounding depends on that required by your job

Find the Perimeter of a Trapezoid Step 19

Step 7. Calculate the numerical value of the hypotenuse of triangle BCH2.

By taking the previous trigonometric formula, you get BC = BH2 / sin α2.

With the numerical data, you get BC = 6 / sin 35 = 10, 46. The side [BC] of the trapezoid therefore measures 10, 5 cm (rounded to the nearest tenth)

Find the Perimeter of a Trapezoid Step 20

Step 8. Do the same for the ADH triangle.1.

The sine of the angle in D, noted α1, corresponds to the ratio between the lengths of the opposite side [AH1] and the hypotenuse [AD]. By taking the trigonometric formula, you get sin α1 = AH1 / AD or AD = AH1 / sin α1.

Suppose the angle at D is 45 °. So you have AD = 6 / sin 45 = 8, 49. The side [AD] of the trapezoid measures 8.5 cm (rounded to the nearest tenth)

Find the Perimeter of a Trapezoid Step 21

Step 9. Calculate the length of the base of the triangles

This will allow you to determine the length of the large base of the trapezoid. Thanks to your previous calculations, you can use the Pythagorean theorem. In a right triangle, the length ratio is particular. According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Note that, by misuse of language, the length of one side is referred to as said side.

Find the Perimeter of a Trapezoid Step 22

Step 10. Calculate the length of the base of triangle BCH2.

This is the segment [CH2]. Using the formula of the Pythagorean theorem and the previous notations, BC2 = BH22 + CH22. By isolating the length of the base, you get CH2 = √ (BC2 - BH22).

  • With digital data you have CH2 = √(10, 52 - 62) = 8, 56. The segment [CH2] measures 8.6 cm (rounded to the nearest tenth).
Find the Perimeter of a Trapezoid Step 23

Step 11. Do the same for the triangle ADH1.

By applying the formula of the Pythagorean theorem, you have AD2 = AH12 + DH12. By isolating the length of the base you get DH1 = √ (AD2 - AH12).

  • Including the numerical data, the length of the base of the triangle is 6 cm. The calculation is DH1 = √(8, 52 - 62) = 6, 02. The segment [DH1] therefore measures 6 cm (rounded to the nearest tenth).
Find the Perimeter of a Trapezoid Step 24

Step 12. Calculate the length of the large base

It is the sum of the length of the rectangle and the bases of the triangles you calculated. By taking the previous notations, you have DC = DH1 + H1H2 + CH2.

By integrating the numerical data, you get DC = 6 + 6 + 8, 6 = 20, 6. The large trapezoid base measures 20.6 cm

Find the Perimeter of a Trapezoid Step 25

Step 13. Calculate the perimeter of the trapezoid

Sum the dimensions of the sides of the polygon. The formula for the perimeter P is as follows: P = AB + BC + CD + DA.

By replacing with the numerical values, you get P = 6 + 10, 5 + 20, 6 + 8, 5 = 45, 6. The perimeter of the trapezoid is therefore 45, 6 cm

Advice

  • Use the properties of the particular triangles to find the missing dimensions. For example, know that the relationships between the lengths are remarkable in the case of isosceles right triangles or right triangles having angles of 30 ° and 60 °.
  • Keep a scientific calculator handy. It is essential for trigonometric calculations and the conversion of square roots. You can also find them online.
  • Indicate the dimensions directly on your figure as you go.
  • Unless your calculations require absolute precision, you can just round the dimensions to the nearest tenth.

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