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How to Calculate the Area of a Triangle

3/30/2014

Though the most common way to find the area of a triangle is to multiply the base and height and to divide the result by two, there are a number of other ways to find the area of a triangle depending on what dimensions you are given. There are other formulas for finding the area of a triangle depending on whether you know the length of the three sides, the length of one side of an equilateral triangle, or the length of two sides and their included angle. '


EditSteps


EditCalculation Help





Area of a Triangle Base Times Height Cheat Sheet






Area of a Triangle Side Length Cheat Sheet






Area of an Equilateral Triangle Cheat Sheet






EditUsing the Base and Height



  1. Find the base and height of your triangle. The base of the triangle is the length of one side, which is usually the bottom side of the triangle. The height is the length from the base to the top corner of the triangle, which is perpendicular to the base. In a right triangle, the base and height are the two sides that join to form a ninety-degree angle. However, in a non-right triangle, such as the one below, the height will cut across the middle of the shape.




    • Once you identify the base and height of the triangle, you can begin to use the formula.



  2. Write down the formula for finding the area of a triangle. The formula for this type of problem is Area = 1/2(base x height), or 1/2(bh). Once you write it down, you can begin to plug in the lengths of the height and base.[1]



  3. Plug in the values for base and height. Identify the base and height of your triangle and put these numbers into the equation. In this example, the height of the triangle is 3 cm and the base of the triangle is 5 cm. This is what the formula would look like after you plug in these values:




    • Area = 1/2 x (3 cm x 5 cm)



  4. Solve the equation. You can multiply the height times the base first since those numbers are in parenthesis and then multiply the result by 1/2, but know that multiplying these three numbers in any order will yield the same result. Just remember to state your answer in square units since you're working with a two-dimensional space. Here is how you solve to get the final answer:




    • Area = 1/2 x (3 cm x 5 cm)

    • Area = 1/2 x 15 cm2

    • Area = 7.5 cm2




EditUsing the Length of Each Side (Heron's Formula)



  1. Calculate the semi-perimeter of the triangle. To find the semi-perimeter of the triangle, all you have to do is add up all of the sides and divide the result by two. The formula for finding the semi-perimeter of a triangle is the following: semiperimeter = (length of side a + length of side b + length of side c) / 2, or s = (a + b + c) / 2. Since you know the three lengths of the right triangle are 3 cm, 4 cm, and 5 cm, you can plug them into the formula and solve for the semi-perimeter:




    • s = (3 + 4 + 5)/2

    • s = 12/2

    • s = 6



  2. Plug the appropriate values into the formula for finding the area of a triangle. The formula for finding the area of a triangle is called Heron's formula and is the following: Area = √{s (s - a)(s - b)(s - c)}. Remember from the previous step that s is the semiperimeter and a, b, and c are the triangle’s three sides. Using the “order of operations,” start by solving everything inside the parenthesis, then everything within the square root, and finally the square root itself. Here is what the formula would look like after you plug in the known values:[2]




    • Area = √{6 (6 - 3)(6 - 4)(6 - 5)}



  3. Subtract the values in the three sets of parenthesis. To do this, simply subtract 6 - 3, 6 - 4, and 6 - 5. Here's what it would look like:




    • 6 - 3 = 3

    • 6 - 4 = 2

    • 6 - 5 = 1

    • Area = √{6 (3)(2)(1)}



  4. Multiply the results of all three sets of parentheses together. Simply multiply 3 x 2 x 1 to get 6. You need to multiply these numbers together before multiplying them by 6 because they are in parenthesis.



  5. Multiply the previous result by the semi-perimeter. Now, multiply the result, 6, by the semi-perimeter, which is also 6. 6 x 6 = 36.



  6. Find the square root. 36 is a perfect square, and √36 = 6. Don't forget the units you started with -- centimeters. State your final answer in square centimeters. The area of the triangle with the sides 3, 4, and 5 is 6 cm2.




EditUsing One Side of an Equilateral Triangle



  1. Find the side of the equilateral triangle. An equilateral triangle has sides of equal length and equal angles. You'll know that you're working with an equilateral triangle either because you're given this information, or because you know that all of the angles or all of the sides have the same value. The value of one side of this equilateral triangle is 6 cm. Write it down.




    • If you know you're working with an equilateral triangle but only know the perimeter, just divide it by 3. For example, the length of a side of an equilateral triangle with the perimeter 9 is simply 9/3, or 3.



  2. Write down the formula used for finding the area of an equilateral triangle. The formula for this type of problem is area = (s^2)(√3)/4. Note that s denotes “side.”[3]





  3. Plug the value of one side into the equation. First, square the value of the side, 6, to get 36. Then, find the decimal value of √3, if your answer must be in decimal form. Just plug √3 into your calculator to get 1.732. Then, divide this number by 4. Note that you could also divide 36 by 4 and then multiply it by √3 -- the order of operations won't affect the answer.



  4. Solve. Now, you just have to do the math. 36 x √3/4 = 36 x .433 = 15.59 cm2 The area of an equilateral triangle with a side 6 cm long is 15.59 cm2.




EditUsing the Lengths of Two Sides and the Included Angle



  1. Find the value of the lengths of two sides and the included angle. The included angle is the angle between the two known sides of the triangle. You must know these values to find the area of a triangle using this method. Let's say you're working with a triangle with the following dimensions:

    • angle A = 123ยบ

    • side b = 150 cm

    • side c = 231 cm



  2. Write down the formula for finding the area of the triangle. The formula for finding the area of a triangle with two known sides and a known included angle is the following: Area = 1/2(b)(c) x sinA. In this equation, "b" and "c" represent the lengths of the sides and "A" represents the angle. You should always take the sine of the angle in this equation.[4]

  3. Plug the values into the equation. Here's what the equation would look like once you plug in the values:




    • Area = 1/2(b)(c) x sinA

    • Area = 1/2(150)(231) x sinA



  4. Solve. To solve this equation, first multiply the sides and divide the result by two, and then multiply this result by the sine of the angle. You can find the value of the sine of the angle by plugging it into the "sin" function on your calculator. Don't forget to state your answer in cubic units. Here's how you do it:




    • Area = 1/2(150)(231) x sinA

    • Area = 1/2(34,650) x sinA

    • Area = 17,325 x sinA

    • Area = 17,325 x .8386705

    • Area = 14,530 cm2




EditTips



  • If you're not exactly sure why the base-height formula works this way, here's a quick explanation. If you make a second, identical triangle and fit the two copies together, it will either form a rectangle (two right triangles) or a parallelogram (two non-right triangles). To find the area of a rectangle or parallelogram, simply multiply base by height. Since a triangle is half of a rectangle or parallelogram, you must therefore solve for half of base times height.




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EditSources and Citations




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