Difference between revisions of "Find the Perimeter of a Trapezoid"
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==Steps== | ==Steps== | ||
===If You Know the Length of Both Sides and Bases=== | ===If You Know the Length of Both Sides and Bases=== | ||
| − | #Set up the formula for perimeter of a trapezoid. The formula is <math>P = T + B + L + R</math>, where <math>P</math> equals the perimeter of the trapezoid, and the variables <math>T</math> equals the length of the top base of the trapezoid, <math>B</math> equals the length of the bottom base, <math>L</math> equals the length of the left side, and <math>R</math> equals the length of the right side.<ref>http://www.mathopenref.com/trapezoidperimeter.html</ref> | + | #Set up the formula for perimeter of a trapezoid. The formula is <math>P = T + B + L + R</math>, where <math>P</math> equals the perimeter of the trapezoid, and the variables <math>T</math> equals the length of the top base of the trapezoid, <math>B</math> equals the length of the bottom base, <math>L</math> equals the length of the left side, and <math>R</math> equals the length of the right side.<ref name="rf1">http://www.mathopenref.com/trapezoidperimeter.html</ref> |
#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. | #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:<br><math>P = 2 + 3 + 1 + 1</math> | #*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:<br><math>P = 2 + 3 + 1 + 1</math> | ||
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#Divide the trapezoid into a rectangle and two right triangles. To do this, draw the height from both top vertices. | #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. | #*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. | ||
| − | #Label each height line. Since these are opposite sides of a rectangle, they will be the same length.<ref>http://www.mathopenref.com/coordrectangle.html</ref> | + | #Label each height line. Since these are opposite sides of a rectangle, they will be the same length.<ref name="rf2">http://www.mathopenref.com/coordrectangle.html</ref> |
#*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. | #*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. | ||
| − | #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.<ref | + | #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.<ref name="rf2" /> 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. | #*For example, if the top base of the trapezoid is 6 cm, then the middle section of the bottom base is also 6 cm. | ||
| − | #Set up the Pythagorean Theorem formula for the first right triangle. The formula is <math>a^{2} + b^{2} = c^{2}</math>, where <math>c</math> is the length of the hypotenuse of the right triangle (the side opposite the right angle), <math>a</math> is the height of the right triangle, and <math>b</math> is the length of the base of the triangle.<ref>http://mathworld.wolfram.com/PythagoreanTheorem.html</ref> | + | #Set up the Pythagorean Theorem formula for the first right triangle. The formula is <math>a^{2} + b^{2} = c^{2}</math>, where <math>c</math> is the length of the hypotenuse of the right triangle (the side opposite the right angle), <math>a</math> is the height of the right triangle, and <math>b</math> is the length of the base of the triangle.<ref name="rf3">http://mathworld.wolfram.com/PythagoreanTheorem.html</ref> |
#Plug the known values from the first triangle into the formula. Make sure you plug in the side length of the trapezoid for <math>c</math>. Plug in the height of the trapezoid for <math>a</math>. | #Plug the known values from the first triangle into the formula. Make sure you plug in the side length of the trapezoid for <math>c</math>. Plug in the height of the trapezoid for <math>a</math>. | ||
#*For example, if you know the height of the trapezoid is 6 cm, and the length of the side (hypotenuse) is 9 cm, your equation will look like this:<br><math>6^{2} + b^{2} = 9^{2}</math> | #*For example, if you know the height of the trapezoid is 6 cm, and the length of the side (hypotenuse) is 9 cm, your equation will look like this:<br><math>6^{2} + b^{2} = 9^{2}</math> | ||
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#Take the square root to find the value of <math>b</math>. (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. | #Take the square root to find the value of <math>b</math>. (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. | ||
#*For example:<br><math>b^{2} = 45</math><br><math>b = \sqrt{45}</math><br><math>b = \sqrt{45}</math><br><math>b = 3\sqrt{5}</math><br>So, you should label <math>3\sqrt{5}</math> on the base of your first triangle. | #*For example:<br><math>b^{2} = 45</math><br><math>b = \sqrt{45}</math><br><math>b = \sqrt{45}</math><br><math>b = 3\sqrt{5}</math><br>So, you should label <math>3\sqrt{5}</math> on the base of your first triangle. | ||
| − | #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,<ref>http://www.mathsisfun.com/geometry/trapezoid.html</ref> the two right triangles are congruent, so you can simply carry the value from the first triangle over to the second triangle. | + | #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,<ref name="rf4">http://www.mathsisfun.com/geometry/trapezoid.html</ref> the two right triangles are congruent, so you can simply carry the value from the first triangle over to the second triangle. |
#*For example, if the second side of the trapezoid is 7 cm, you would calculate:<br><math>a^{2} + b^{2} = c^{2}</math><br><math>6^{2} + b^{2} = 7^{2}</math><br><math>36 + b^{2} = 49</math><br><math>b^{2} = 13</math><br><math>b = \sqrt{13}</math><br>So, you should label <math>\sqrt{13}</math> on the base of your second triangle. | #*For example, if the second side of the trapezoid is 7 cm, you would calculate:<br><math>a^{2} + b^{2} = c^{2}</math><br><math>6^{2} + b^{2} = 7^{2}</math><br><math>36 + b^{2} = 49</math><br><math>b^{2} = 13</math><br><math>b = \sqrt{13}</math><br>So, you should label <math>\sqrt{13}</math> 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: <math>P = T + B + L + R</math>. 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. | #Add up all the side lengths of the trapezoid. The perimeter of any polygon is the sum of all sides: <math>P = T + B + L + R</math>. 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. | ||
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#Divide the trapezoid into a rectangle and two right triangles. To do this, draw the height from both top vertices. | #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. | #*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. | ||
| − | #Label each height line. Since these are opposite sides of a rectangle, they will be the same length.<ref | + | #Label each height line. Since these are opposite sides of a rectangle, they will be the same length.<ref name="rf2" /> |
#*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. | #*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. | ||
| − | #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.<ref | + | #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.<ref name="rf2" /> |
#*For example, if the top base of the trapezoid is 6 cm, then the middle section of the bottom base is also 6 cm. | #*For example, if the top base of the trapezoid is 6 cm, then the middle section of the bottom base is also 6 cm. | ||
#Set up the sine ratio for the first right triangle. The ratio is <math>\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}</math>, where <math>\theta</math> is the measure of the interior angle, <math>\text{opposite}</math> is the height of the triangle, and <math>\text{hypotenuse}</math> is the length of the hypotenuse. | #Set up the sine ratio for the first right triangle. The ratio is <math>\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}</math>, where <math>\theta</math> is the measure of the interior angle, <math>\text{opposite}</math> is the height of the triangle, and <math>\text{hypotenuse}</math> is the length of the hypotenuse. | ||
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== Tips == | == Tips == | ||
| − | *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,<ref>http://www.regentsprep.org/regents/math/algtrig/att2/ltri30.htm</ref> or a 90-45-45 triangle.<ref>http://www.regentsprep.org/regents/math/algtrig/att2/ltri45.htm</ref> | + | *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,<ref name="rf5">http://www.regentsprep.org/regents/math/algtrig/att2/ltri30.htm</ref> or a 90-45-45 triangle.<ref name="rf6">http://www.regentsprep.org/regents/math/algtrig/att2/ltri45.htm</ref> |
| − | *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.<ref>http://www.csuchico.edu/~jhudson/pdf/trigtabl.pdf</ref> | + | *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.<ref name="rf7">http://www.csuchico.edu/~jhudson/pdf/trigtabl.pdf</ref> |
==Things You'll Need== | ==Things You'll Need== | ||
