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How Do You Find the Area of a Quarter Circle?

area of a quarter circle: Image of a blue circle with a quarter of it separated and colored red

Finding the perimeter and area of a quarter circle is a good exercise in working with the two circle formulas you should be aware of: the formulas for the perimeter and area. Let's review these two formulas:

Circumference = 2ℼr

Area = ℼr²

Finding the Area of a Quarter Circle

area of a quarter circle: Diagram showing the equation of the area of a circle inside a circle

Finding the area of a quarter circle starts with finding the area of a whole circle. We know that the equation of the area of the whole circle is A = ℼr².

If we want to find the area of a semicircle, we would divide that area by two. That makes the following equation:

area of a quarter circle: Formula for the area of a semicircle

If we want to further break it down into the area of a quadrant, we know that each quadrant is equivalent to:

area of a quarter circle: Formula for the area of a quadrant

And that’s it! The area of a quarter circle is exactly one-fourth the area of a full circle.

Finding the Perimeter of a Quarter Circle

Diagram showing the perimeter of a quarter circle

The perimeter of a quarter circle is a little trickier. It includes two segments that are the radius of the quarter circle, added with the curved part that is the outside of the circle.

We know that the perimeter of a circle is equal to the circumference of a circle, or C = 2ℼr. So, we know that the curved part to the quarter circle is equivalent to one-fourth of that, or:

Formula for the curved part of the quarter circle

Now for the two straight edges. We know that the radius of a quarter circle is the same as the radius of the whole circle. That is, it is equal to the radius r that was used in the formula for the circumference. And we have to add two sides.

Formula for the perimeter of a quarter circle

Finding the Perimeter and Area of a Quarter Circle

In either case, finding the perimeter or the area of the quarter circle starts with knowing the circle formulas themselves.

Circumference = 2ℼr

Area = ℼr²

To find the area of a quarter circle, just divide the area of the full circle by four. For perimeter, don’t forget that you have to account for both the outside curve, as well as the two inner segments.

More Math Homework Help:

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