Evaluate the Double Integral Xy Da Where D Is the Triangular Region With Vertices

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The most common way to find the area of a triangle is to take half of the base times the height. Numerous other formulas exist, however, for finding the area of a triangle, depending on what information you know. Using information about the sides and angles of a triangle, it is possible to calculate the area without knowing the height.

1. 1

   Find the base and height of the triangle. The base is one side of the triangle. The height is the measure of the tallest point on a triangle. It is found by drawing a perpendicular line from the base to the opposite vertex. This information should be given to you, or you should be able to measure the lengths.

   • For example, you might have a triangle with a base measuring 5 cm long, and a height measuring 3 cm long.

2. 2

   Set up the formula for the area of a triangle. The formula is ${\displaystyle {\text{Area}}={\frac {1}{2}}(bh)}$ , where ${\displaystyle b}$ is the length of the triangle's base, and ${\displaystyle h}$ is the height of the triangle.^[1]

3. 3

   Plug the base and height into the formula. Multiply the two values together, then multiply their product by ${\displaystyle {\frac {1}{2}}}$ . This will give you the area of the triangle in square units.

4. 4

   Find the area of a right triangle. Since two sides of a right triangle are perpendicular, one of the perpendicular sides will be the height of the triangle. The other side will be the base. So, even if the height and/or base is unstated, you are given them if you know the side lengths. Thus you can use the ${\displaystyle {\text{Area}}={\frac {1}{2}}(bh)}$ formula to find the area.

1. 1

   Calculate the semiperimeter of the triangle. The semi-perimeter of a figure is equal to half its perimeter. To find the semiperimeter, first calculate the perimeter of a triangle by adding up the length of its three sides. Then, multiply by ${\displaystyle {\frac {1}{2}}}$ .^[2]

2. 2

   Set up Heron's formula. The formula is ${\displaystyle {\text{Area}}={\sqrt {s(s-a)(s-b)(s-c)}}}$ , where ${\displaystyle s}$ is the semiperimeter of the triangle, and ${\displaystyle a}$ , ${\displaystyle b}$ , and ${\displaystyle c}$ are the side lengths of the triangle.^[3]

3. 3

   Plug the semiperimeter and side lengths into the formula. Make sure you substitute the semiperimeter for each instance of ${\displaystyle s}$ in the formula.

4. 4

   Calculate the values in parentheses. Subtract the length of each side from the semiperimeter. Then, multiply these three values together.

5. 5

   Multiply the two values under the radical sign. Then, find their square root. This will give you the area of the triangle in square units.

1. 1

   Find the length of one side of the triangle. An equilateral triangle has three equal side lengths and three equal angle measurements, so if you know the length of one side, you know the length of all three sides.^[4]

   • For example, you might have a triangle with three sides that are 6 cm long.

2. 2

   Set up the formula for the area of an equilateral triangle. The formula is ${\displaystyle {\text{Area}}=(s^{2}){\frac {\sqrt {3}}{4}}}$ , where ${\displaystyle s}$ equals the length of one side of the equilateral triangle.^[5]

3. 3

   Plug the side length into the formula. Make sure you substitute for the variable ${\displaystyle s}$ , and then square the value.

4. 4

   Multiply the square by $$ 3 {\displaystyle {\sqrt {3}}} . It's best to use the square root function on your calculator for a more precise answer. Otherwise, you can use 1.732 for the rounded value of ${\displaystyle {\sqrt {3}}}$ .

5. 5

   Divide the product by 4. This will give you the area of the triangle in square units.

1. 1

   Find the length of two adjacent sides and the included angle. Adjacent sides are two sides of a triangle that meet at a vertex.^[6] The included angle is the angle between these two sides.

   • For example, you might have a triangle with two adjacent sides measuring 150 cm and 231 cm in length. The angle between them is 123 degrees.

2. 2

   Set up the trigonometry formula for the area of a triangle. The formula is ${\displaystyle {\text{Area}}={\frac {bc}{2}}\sin A}$ , where ${\displaystyle b}$ and ${\displaystyle c}$ are the adjacent sides of the triangle, and ${\displaystyle A}$ is the angle between them.^[7]

3. 3

   Plug the side lengths into the formula. Make sure you substitute for the variables ${\displaystyle b}$ and ${\displaystyle c}$ . Multiply their values, then divide by 2.

4. 4

   Plug the sine of the angle into the formula. You can find the sine using a scientific calculator by typing in the angle measurement then hitting the "SIN" button.

5. 5

   Multiply the two values. This will give you the area of the triangle in square units.

Add New Question

• Question

  How do I find the length and width of a triangle before calculating the area?

  

  It should be included in the problem. If it is a right triangle, use the Pythagorean Theorem (A squared + B squared = C squared) to find the missing side.

• Question

  How can I calculate the area of an equilateral triangle?

  

  If you know the base and height, you can use the standard formula A = 1/2bh. If you know the three side lengths, you can use the method for equilateral triangles described in this article.

• Question

  How can I find the area of an isosceles right triangle?

  

  The legs must be the sides that are equal, so you just square the length of one of the legs and divide by 2. If you only have the hypotenuse: since isosceles right triangles come in the ratio 1-1-(square root of 2), you just divide the hypotenuse by sqrt(2), square what you get, and divide by 2.

• Question

  If an equilateral triangle has an x for all sides, what is the area?

  

  This involves trigonometry. You have to find the height of the triangle, which is the distance from one vertex to the opposite side. The height is found by multiplying the length of a side (x) by half the tangent of 60°. (60° represents each of the angles in an equilateral triangle.) Half the tangent of 60° is 0.866. Thus the height is 0.866x. Multiply that by x and divide by two to get the area.

• Question

  A triangle has an area of 24 square units. Its height is 6 units. What is the length of its base?

  

  To find the base, double the area, then divide by the height.

• Question

  A triangle has a base length of 2x+4 and a height of 3y; what is the area?

  

  Without more information, you can't find an exact value. You can, however, state the height as the value of 1/2bh by plugging in these expressions for the base and height. So the area is 1/2(2x+4)(3y); (x+2)(3y); 3xy + 6y.

• Question

  How do I calculate the height of a triangle if I know the area and the base?

  

  Double the area, then divide by the base.

• Question

  How can I mark the center point of any type of triangle?

  

  To find the centroid of a triangle, use the formula from the preceding section that locates a point two-thirds of the distance from the vertex to the midpoint of the opposite side. For example: to find the centroid of a triangle with vertices at (0,0), (12,0) and (3,9), first find the midpoint of one of the sides.

• Question

  If the sides of a triangle are 14 cm, 11 cm, and 9 cm, and the height is 7 cm, what is the area?

  

  Since you know all three side lengths, you can use Heron's formula as described in this article. If you know which side functions as the base, you can use the formula A = 1/2bh

• Question

  How do I calculate area when two angles and one side have been given?

  

  Assuming your triangle is a right triangle, you can use trigonometry to find the other missing side lengths. Once you have all side lengths, you can use Heron's formula, as described in this article, to find the area. For more information on how to use trigonometry, you can read the following article: http://www.wikihow.com/Use-Right-Angled-Trigonometry

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• 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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Article Summary X

To calculate the area of a triangle, start by measuring 1 side of the triangle to get the triangle's base. Then, measure the height of the triangle by measuring from the center of the base to the point directly across from it. Once you have the triangle's height and base, plug them into the formula: area = 1/2(bh), where "b" is the base and "h" is the height. To learn how to calculate the area of a triangle using the lengths of each side, read the article!

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Evaluate the Double Integral Xy Da Where D Is the Triangular Region With Vertices

Source: https://www.wikihow.com/Calculate-the-Area-of-a-Triangle

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