This article was coauthored by wikiHow Staff. Our trained team of editors and researchers validate articles for accuracy and comprehensiveness. wikiHow's Content Management Team carefully monitors the work from our editorial staff to ensure that each article is backed by trusted research and meets our high quality standards.
This article has been viewed 37,703 times.
Learn more...
The area of a closed figure is the space inside measured in square units. For most polygons, like triangles, the area is calculated using the length of the base and the height. Since a circle has no base or height, the area is calculated using the radius. Despite these differences, you can use various methods to create a triangle that has the same area as a given circle, and vice versa.
Steps
Method 1
Method 1 of 3:Using Archimedes' Theorem

1Find the length of the circle's radius. This information should be given, or else you should be able to measure it. If you do not know the radius of the circle, you cannot use this method.
 For example, you might have a circle with a radius of 4 cm.

2Set up the formula for Archimedes' Theorem. This theorem states that the area of any circle is equal to the area of a right triangle whose base is equal to the radius of the circle, and whose height is equal to the circumference of the circle. Mathematically, this is shown by the formula , where is the radius of the circle.^{[1] X Research source }
 Note that is the formula for the area of a circle, and is the formula for the area of a triangle.^{[2] X Research source } The formula is set up to show that the triangle will have a base equal to the radius (), and a height equal to the circumference of a circle ().^{[3] X Research source }
Advertisement 
3Plug the length of the radius into the formula. Make sure you substitute for all three instances of .
 For example, if the radius is 4 cm, the equation will look like this: .

4Calculate the area of the circle. This will also be the area of the triangle. This is shown in the formula by . If you are not using a scientific calculator, use 3.14 as the value of .
 For example:
 So, the area of the circle and the triangle is about 50.24 square centimeters.
 For example:

5Calculate the circumference of the circle. This will give you the height of your triangle. (Remember that the base of the triangle is equal to the radius of the circle). The circumference is shown in the formula by . If you are not using a scientific calculator, use 3.14 as the value of .
 For example:
 So, the height of the triangle is about 25.12 cm.
 For example:

6Check your work. Complete the calculations in the equation to make sure that both sides are equal. Note that if you rounded to 3.14 when using the equation might be a few decimal points off.
 For example:
 Since you rounded to 3.14, and the equation is only off by 2 hundredths, you can assume that the areas are equal, and thus your calculations are correct. Thus, the area of a circle with a radius of 4 cm is equal to the area of a right triangle with a base of 4 cm and a height of 25.12 cm.
Advertisement  For example:
Method 2
Method 2 of 3:Using the Radius of a Circle and the Height of a Triangle

1Set up the formula for the area of a circle. The formula is , where equals the area of the circle and equals the radius of the circle.^{[4] X Research source }

2Plug the length of the radius into the formula and square it. Remember to substitute for variable .
 For example, if the circle has a radius of 4 cm, your formula will look like this:
.
 For example, if the circle has a radius of 4 cm, your formula will look like this:

3Multiply by . If you are not using a calculator, use 3.14 for . This will give you the area of the circle.
 For example:
 So, the area of the circle is about 50.24 cm.
 For example:

4Set up the formula for the area of a triangle. The formula is , where equals the area of the triangle, equals the length of the triangle's base, and equals the height of the triangle.^{[5] X Research source }

5Plug the area into the triangle formula. Since you want the area of each figure to be the same, use the area you previously calculated for the circle.
 For example, if you found the area of the circle to be 50.24 cm, your formula will look like this: .

6Plug the height of the triangle into the formula. You can also use this method if you are given the length of the base (). Just plug the appropriate value in for the corresponding variable.
 For example, if the height of the triangle is 10 cm, your formula will look like this: .

7Multiply the height of the triangle by . Then, divide each side of the equation by this product. This will give you the length of the base of your triangle.
 For example:
 So, the area of a circle with a radius of 4 cm is equal to the area of a triangle with a height of 10 cm and a base of about 10 cm.
Advertisement  For example:
Method 3
Method 3 of 3:Using the Base and Height of a Triangle

1Set up the formula for the area of a triangle. The formula is , where equals the area of the triangle, equals the length of the triangle's base, and equals the height of the triangle.^{[6] X Research source }

2Plug the length of the base and height into the formula. These values should be given to you, or you should be able to measure them.
 For example, if the base of the triangle is 5 cm, and the height of the triangle is 20 cm, then your equation will look like this: .

3Multiply the base and height, then multiply the product by . This will give you the area of the triangle.
 For example:
 So, the area of the triangle is 50 square centimeters.
 For example:

4Set up the formula for the area of a circle. The formula is , where equals the area of the circle and equals the radius of the circle.^{[7] X Research source }

5Plug the area into the circle formula. Since you want the area of each figure to be the same, use the area you previously calculated for the triangle.
 For example, if you found the area of the triangle to be 50 cm, your formula will look like this: .

6Divide each side of the equation by . If you are not using a scientific calculator, you can round to 3.14.
 For example:
 For example:

7Take the square root of each side of the equation. This will give you the length of the radius of a circle with an area equal to that of the triangle.
 For example:
.  So, the area of a circle with a radius of about 4 cm is equal to the area of a triangle with a base of 5 cm and a height of 20 cm.
Advertisement  For example:
Community Q&A
Tips
References
 â†‘ http://www.ams.org/samplings/featurecolumn/fc201202
 â†‘ http://www.mdc.edu/main/images/common_math_formulas_tcm633520.pdf
 â†‘ http://www.mathopenref.com/circumference.html
 â†‘ http://www.mathopenref.com/circlearea.html
 â†‘ https://www.mathsisfun.com/algebra/trigareatrianglewithoutrightangle.html
 â†‘ https://www.mathsisfun.com/algebra/trigareatrianglewithoutrightangle.html
 â†‘ http://www.mathopenref.com/circlearea.html