Math Antics - Perimeter



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How to Find the Perimeter of a Triangle

Three Methods:

Finding the perimeter of a triangle means finding the distance around the triangle.The simplest way to find the perimeter of a triangle is to add up the length of all of its sides, but if you don't know all of the side lengths you will need to calculate them first. This article will first teach you to find the perimeter of a triangle when you do know all three side lengths; this is the easiest and most common way. It will then teach you to find the perimeter of a right triangle when only two of the side lengths are known. Finally, it will teach you to find the perimeter of any triangle for which you know two side lengths and the angle measure between them (an "SAS Triangle"), using the Law of Cosines.

Steps

Finding the Perimeter When Three Side Lengths are Known

  1. Remember the formula for finding the perimeter of a triangle.For a triangle with sidesa,bandc, the perimeterPis defined as:P = a + b + c.
    • What this formula means in simpler terms is that to find the perimeter of a triangle, you just add together the lengths of each of its 3 sides.
  2. Look at your triangle and determine the lengths of the three sides.In this example, the length of sidea=5, the length of sideb=5, and the length of sidec=5.
    • This particular example is called an equilateral triangle, because all three sides are of equal length. But remember that the perimeter formula is the same for any kind of triangle.
  3. Add the three side lengths together to find the perimeter.In this example,5 + 5 + 5 = 15. Therefore,P = 15.
    • In another example, wherea = 4,b = 3, andc=5, the perimeter would be:P = 3 + 4 + 5, or12.
  4. Remember to include the units in your final answer.If the sides of the triangle are measured in centimeters, then your answer should also be in centimeters. If the sides are measured in terms of a variable like x, your answer should also be in terms of x.
    • In this example, the side lengths are each 5cm, so the correct value for the perimeter is 15cm.

Finding the Perimeter of a Right Triangle When Two Sides are Known

  1. Remember what a right triangle is.A right triangle is a triangle that has one right (90 degree) angle. The side of the triangle opposite the right angle is always the longest side, and it is called the hypotenuse. Right triangles show up frequently on math tests, and fortunately there is a very handy formula for finding the length of unknown sides!
  2. Recall the Pythagorean Theorem.The Pythagorean Theorem tells us that for any right triangle with sides of length a and b, and hypotenuse of length c,a2+ b2= c2.
  3. Look at your triangle, and label the sides "a," "b," and "c".Remember that the longest side of the triangle is called the hypotenuse. It will be opposite the right angle and must be labeledc. Label the two shorter sidesaandb. It doesn't really matter which is which, the math will turn out the same!
  4. Enter the side lengths that you know into the Pythagorean Theorem.Remember thata2+ b2= c2. Substitute the side lengths in for the corresponding letters in the equation.
    • If, for example, you know that sidea = 3and sideb = 4, then plug those values into the formula as follows:32+ 42= c2.
    • If you know the length of sidea = 6, and the hypotenusec = 10, then you should set the equation up like so:62+ b2= 102.
  5. Solve the equation to find the missing side length.You will first need to square the known side lengths which means multiplying each value by itself (for example 32= 3 * 3 = 9). If you are looking for the hypotenuse, simply add the two values together and find the square root of this number to find the length. If it is a side length you are missing, you must do a bit of easy subtraction, and then take the square root to get your side length.
    • In the first example, square the values in32+ 42= c2and find that25= c2. Then calculate the square root of 25 to find thatc = 5.
    • In the second example, square the values in62+ b2= 102to find that36 + b2= 100. Subtract 36 from each side to find thatb2= 64, then take the square root of 64 to find thatb = 8.
  6. Add up the lengths of the three side lengths to find the perimeter.Recall that the perimeterP = a + b + c. Now that you know the lengths of sidesa,bandc, you simply need to add the lengths together to find the perimeter.
    • In our first example,P = 3 + 4 + 5, or 12.
    • In our second example,P = 6 + 8 + 10, or 24.

Finding the Perimeter of an SAS Triangle Using the Law of Cosines

  1. Learn the Law of Cosines.The Law of Cosines allows you to solve any triangle when you know two side lengths and measurement of the angle between them. It works on any triangle, and is a very useful formula. The Law of Cosines states that for any triangle with sidesa,b, andc, with opposite anglesA,B, andC:c2= a2+ b2- 2abcos(C).
  2. Look at your triangle and assign variable letters to its components.The first side that you know should be labeleda, and the angle opposite it isA. The second side that you know should be labeledb; the angle opposite it isB. The angle that you know should be labeledC, and the third side, the one you need to solve in order to find the perimeter of the triangle, is sidec.
    • For example, imagine a triangle with side lengths 10 and 12, and an angle between them of 97°. We will assign variables as follows:a = 10,b = 12,C = 97°.
  3. Plug your information into the equation and solve for side c.You will first need to find the squares of a and b, and add them together. Then find the cosine of C using thecosfunction on your calculator, or an online cosine calculator.Multiplycos(C)by2aband subtract the product from the sum ofa2+ b2. The result isc2. Find the square root of this value and you have the length of sidec. Using our example triangle:
    • c2= 102+ 122- 2 × 10 × 12 ×cos(97).
    • c2= 100 + 144 – (240 × -0.12187)(Round the cosine to 5 decimal places.)
    • c2= 244 – (-29.25)
    • c2= 244 + 29.25(Carry the minus symbol through whencos(C) is negative!)
    • c2= 273.25
    • c = 16.53
  4. Use side lengthcto find the perimeter of the triangle.Recall that PerimeterP = a + b + c, so all you need to do is add the length you just calculated for sidecto the values you already had foraandb.
    • In our example:10 + 12 + 16.53 = 38.53, the perimeter of our triangle!

Community Q&A

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  • Question
    Can I solve the perimeter of a triangle if I'm only given three included angles?
    Orangejews
    Community Answer
    No. You can know the exact shape of the triangle from the angles but nothing about its size, so the perimeter could be anything. (At least for Euclidean geometry; in spherical geometry, all similar triangles are congruent so you do know the scale and thus the perimeter.)
    Thanks!
  • Question
    How do I work out three sides of a triangle when I know the perimeter is 52 meters, the longest side is 12 meters longer than the shortest and the third side is twice the length of the shortest side?
    Top Answerer
    S + (S + 12) + (S x 2) = 52.
    Thanks!
  • Question
    Two sides of a triangle are each 7 cm long. The length of the third side is an integer number of centimeters. At most how many centimeters do the perimeter of the triangle measure?
    Orangejews
    Community Answer
    Triangle inequality says no side can be longer than the sum of the other two. If the third side were 14 cm, it would make a line segment, not a triangle, but the next integer smaller 13 cm works. That makes a perimeter of 27 cm.
    Thanks!
  • Question
    How do I find the perimeter of a circle?
    Top Answerer
    Diameter multiplied by pi.
    Thanks!
  • Question
    How do I determine the perimeter of a triangle when I only know the base and height?
    wikiHow Contributor
    Community Answer
    You can't find the perimeter with just the base and height because you need the measure of at least two sides and an angle between them.
    Thanks!
  • Question
    The base of an isosceles triangle is 4/3 cm. The perimeter of the triangle is 42/15 cm. What is the length of either of the remaining equal sides?
    Top Answerer
    Subtract 4/3 from 42/15, then divide by 2.
    Thanks!
  • Question
    If an angle is 5 degrees and two sides are 10 feet, what is the formula to figure out the third side?
    Top Answerer
    It depends on where the angle is situated in relation to the two sides.
    Thanks!
  • Question
    One side of triangle is 13 cm and another side is 17 cm, so what can be the perimeter of triangle?
    Orangejews
    Community Answer
    The triangle inequality means the third side must be between 4 and 30cm, so the perimeter is between 34 and 60 cm.
    Thanks!
  • Question
    If I fold a square in half to make a triangle, what is the perimeter?
    wikiHow Contributor
    Community Answer
    You have to use the base and length, and then measure the slant, or use the Pythagorean Theorem to find the slant.
    Thanks!
  • Question
    How can I find the perimeter of a triangle when only given the vertices of the triangle on a coordinate plane?
    Top Answerer
    Calculate the length of each side using the Pythagorean theorem, and add them together.
    Thanks!
Unanswered Questions
  • How can I find the length of a side if I only know the value of p and the size of two angles?
  • How do you find the perimeter of a triangle?
  • If the perimeter of the triangle above is 15, what is the measure of angle x?
  • How do I find the length of third side of a triangle when I know 2 of the sides?
  • How do I find the perimeter of a right angled triangle if the opposite side is 5cm and we have a degree of 30 degrees?
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Quick Summary

To find the perimeter of a triangle, add up the lengths of all of the sides. If you don't have the length of each side, you'll need to find the length of the missing side in order to find the perimeter. For a right triangle, you would find the length of the missing side using the formula a^2 + b^2 = c^2, also known as the Pythagorean Theorem. If you're working with a triangle that doesn't have a right angle, use the Law of Cosines to find the length of the missing side, which is c^2 = a^2 + b^2 - 2ab cos(C).

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Date: 11.12.2018, 08:14 / Views: 34291