 # Area of Circles, Sectors and Segments

A series of free, online High School Geometry Video Lessons.
Videos, worksheets, and activities to help Geometry students.

In this lesson, we will learn

• how to calculate the area of a circle
• how to calculate the area of a sector
• how to calculate the area of a segment
• how to calculate the area of an annulus
• how to calculate the area of the region between a circle and a square (Inscribed circle)

Related Topics:
More lessons for High School Geometry, More Geometry Lessons

### Area of Circles

The area of circles is derived by dividing a circle into an infinite number of wedges formed by radii drawn from the center. When these wedges are rearranged, they form a rectangle whose height is the radius of the circle and whose base length is half of the circumference of the circle. The area of circles are also used in sectors, segments and annuluses.

How to derive the formula to calculate the area of a circle?
A = πr2 How to find the area and circumference of a circle?
Example:
Find the circumference and area of a circle with radius 8m.

### Area of a Sector

A sector in a circle is the region bound by two radii and the circle. Since it is a fractional part of the circle, the area of any sector is found by multiplying the area of the circle, π × r2, by the fraction x/360, where x is the measure of the central angle formed by the two radii. The area of a sector is also used in finding the area of a segment.

How to derive the formula to calculate the area of a sector in a circle?
Two examples are shown: one for finding the sector of a circle and the other to find the radius of a circle with a given sector area.
Examples:
1. Find the area of the shaded region.
2. Find the radius of the circle is the area of the shaded region is 50π. How to find the area of a sector?

### Area of a Segment

The area of a segment in a circle is found by first calculating the area of the sector formed by the two radii and then subtracting the area of the triangle formed by the two radii and chord (or secant).
In segment problems, the most challenging aspect is often calculating the area of the triangle. Related topics include area of a sector, area of a circle and area of an annulus.

How to calculate the area of segments of circles?
Segments are regions bounded by an arc of a circle and a chord (line segment).
Introduction to area of sectors and triangles are prerequisite to this video.
Examples:
1. Determine the area of the blue segment.
2. Determine the area of the red segment.
3. The area of the orange region is approximately 177.82. Determine θ to the nearest degree.
How to derive the area of a segment formula?
Area of segment = Area of sector - area of triangle

### Area of an Annulus

An annulus is similar to a ring or a castle's moat; it is the area between two concentric circles. Calculating the annulus area, therefore, involves finding the difference of the two circles' area.
A common trick on annulus problems is to give the distance between the small and large circle, and not the large circle's radius. Related topics include area of a sector, area of a circle and area of an segment.

How to derive the formula for the area of an annulus?
Area of annulus = Area of large circle - Area of small circle
Example:
Find the area of the annulus, correct to 1 decimal place.

### Regions Between Circles and Squares

A common application of the area of a circle and the area of a square are problems where a circle is circumscribed about a square or inscribed in a square. Regions between circles and squares problems almost always involve subtracting the two areas; their difficulty stems from dimensions given for one but not both shapes. Related topics include area of sectors and area of circles.

How to calculate the area between a square and an inscribed circle?
Example:
In the figure, a circle is inscribed in a square.
a. Find the area of the circle.
b. Find the area of the shaded region.

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