Geometry B Assignment 13: Area of Circles Segments

A chord divides a circle into two segments.

Given a circle with an 8" radius, find the area of the smaller segment whose chord is 8" long.

Find the area of the larger segment whose chord is 8" long in a circle with an 8" radius.

A segment of a circle has a 120° arc and a chord of 8√3 in. Find the area of the segment.

Find the area of one segment formed by a square with sides of 6" inscribed in a circle.

Find the area of a segment formed by a side of a regular hexagon with sides of 3" inscribed in a circle.

An equilateral triangle is inscribed in a circle with a radius of 6". Find the area of the segment cut off by one side of the triangle.

Find the area of the shaded portion in the equilateral triangle with sides equal to 6.

Find the area of the shaded portion in the square. (Assuming the central point of the arc is the corresponding corner)

Find the area of the shaded portion in the square. (Assuming the central point of each arc is the corresponding central point of the line)

Find the area of the shaded portion in the square. (Assuming the central point of each arc is the corresponding corner)

Find the area of the shaded portion in the circle.

Find the area of the shaded portion intersecting between the two circles.

Find the area of the shaded portion.

Chords and Segments

A chord of a circle creates two segments within that circle.

Area Calculations for Segments

For a circle with an 8" radius and an 8" long chord, the area of the smaller segment is ( A = \frac{32}{3} \pi - 16 \sqrt{3} ) in².
The area of the larger segment corresponding to the same chord and radius is ( A = \frac{160}{3} \pi + 16 \sqrt{3} ) in².

Angled Segments

A circle segment with a 120° arc and a chord of ( 8\sqrt{3} ) in has an area of ( A = \frac{64}{3} \pi - 16 \sqrt{3} ) in².

Inscribed Segment Areas

For a square with 6" sides inscribed in a circle, the area of one resulting segment is ( A = \frac{9}{2} \pi - 9 ) in², using a radius ratio of 1:1:√2.
A regular hexagon inscribed in a circle with side lengths of 3" yields a segment area of ( A = \frac{3}{2} \pi - \frac{9}{4} \sqrt{3} ) in², with the area of an equilateral triangle utilized.

Equilateral Triangle Segments

An equilateral triangle inscribed in a circle with a 6" radius provides a segment area cut off by one side, calculated as ( A = 12 \pi - 9 \sqrt{3} ) in².

Shaded Areas in Geometric Shapes

The shaded area within an equilateral triangle with sides of 6" is given by ( A = 9 \sqrt{3} - \frac{9}{2} \pi ).
Within a square, the shaded area from corners results in ( A = 36 - 9 \pi ) in².
When the central points of arcs are at the corresponding line midpoint of a square, the shaded area calculates to ( A = \pi - 2 ).
If the arcs originate at the square corners, the shaded area is ( A = 8 - 2 \pi ).

Circle Intersections

The shaded area from an intersection of two circles is expressed as ( A = \frac{16}{3} \pi - 8 \sqrt{3} ).
The area of the shaded portion within a single circle equals ( A = 30 \pi - 9 \sqrt{3} ).

Overall Formulas

Formulas often involve π terms and square root components illustrating the geometry of the shapes involved, addressing both circular and polygonal contexts.

Test your knowledge on the area of circle segments in Geometry B. This quiz covers definitions and calculations related to chords and segments within circles, including exact area expressions. Perfect for assignment preparation and review.

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Geometry B Assignment 13: Area of Circles Segments

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