How to Calculate the Area of an Arch

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An arch consists of a rectangle with a sector of a circle on top of it. To calculate the area of an arch you must first calculate the area of the sector, then add the area of the rectangle to it. A sector of a circle is an interior portion of a circle bounded by two radii and an arc. An arc of a circle is any connected section of the circle's circumference. Arcs can be measured by the central angle, created by drawing radii from the endpoints of the arc, or by the length of the arc itself. The area of a sector can be calculated using the measure of the central angle, in radians.

Things You'll Need

  • Protractor
  • Ruler
  • Draw a line from one endpoint of the arch's curve to the other endpoint of the arches curve. This line will divide the arch into a half circle and a rectangle.

  • Measure the length of the line and record this length for use later.

  • Divide the length of the line by 2 to find the radius of the circle. Record this value for later use.

  • Plug the radius of the circle into the equation A = (1/2)?r^2 for the variable r. Because we are measuring the area of a half circle; ?, the central angle, is equal to ? radians.

  • Calculate the value of A and record the results for later use. This is the area of the sector at the top of the arch.

  • Measure the distance from the bottom of the arch to the line drawn in Step 1.

  • Multiply the distance measured in Step 6 by the length of the line measured in Step 2 to determine the area of the rectangle.

  • Add the area of the rectangle, from step 7, to the area of the half circle, from step 5, to determine the total area of the arch.

References

  • Photo Credit Arch image by Semfamily from Fotolia.com
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