Geometry 6.8 Area and sectors

there are three things you need to do to calculate the area of a sector

1. Find the area of the whole circle that contains the sector. 2. Find the fraction of the circle covered by the sector. 3. Multiply the area by the fraction.

semicircle:

A 180° arc; half of a circle.

circle

A geometric figure consisting of all the points on a plane that are the same distance from a single point, called its center.

secant

A line or a line segment that intersects a circle in two points.

diameter

A line segment that contains the center of the circle and has endpoints on the circle. This term also refers to the length of this line segment; the ______________of a circle is twice the radius.

radius

A line segment that has one endpoint at the center of a circle and the other endpoint on the circle. ___________ also means "the length of such a line segment." The _____________of a circle is half its diameter. The plural of _______________ is radii.

tangent line

A line that intersects a circle at exactly one point, known as the point of tangency.

vertical angles

A pair of opposite angles formed by intersecting lines. Vertical angles have equal measures.

intercepted arc

A part of the circle (an arc) that is cut off from the rest of the circle's circumference by lines or segments intersecting the circle.

arc

A part of the circumference of a circle.

Sector

A part of the interior of a circle bounded by an arc and two radii that share the arcs endpoints.

What is special about a radius that is perpendicular to a chord?

A radius that is perpendicular to the chord divides the chord into two equal pieces

Radian

A unit of angular measure determined by the condition: The central angle of one radian in a circle of radius 1 produces an arc of length 1.

tangent-chord angle

An angle formed by a tangent and a chord that shares the point of tangency. The measure of a tangent-chord angle is half the measure of the intercepted arc inside the angle.

inscribed angle

An angle formed by two chords of a circle that share an endpoint. It is not a central angle

central angle

An angle that has its vertex at the center of a circle.

major arc

An arc of a circle that is longer than half the circumference. The degree measure of a ____________ arc is greater than 180°.

minor arc

An arc of a circle that is shorter than half the circumference. The degree measure of a _______arc is less than 180°.

chord

Any line segment whose endpoints are on the circle.

What is special about chords that are the same distance from the center?

Chords that are the same distance from the center of the circle have the same length.

How are the angles formed by the chords and the intercepted arcs related?

Intersecting chords form a pair of congruent vertical angles. Each angle measure is half the sum of the intercepted arcs.

secant-secant angle

The angle formed by the intersection of two secants of the same circle. The measure of a secant-secant angle is half the difference of the measures of the intercepted arcs.

Describe how the area of a full circle compares to the area of slives of that circle.

The areas of the slices together form the area of the entire circle.

circumference

The distance around a circle

How do you find the distance between the center point and a chord?

The distance is the measure of the perpendicular bisector between the center point and the chord.

What is the relationship between the circle's height and circumference?

The length of the circumference is roughly three times the circle's height.

secant-secant theorem

The measure of a secant-secant angle is one-half of the difference between the two intercepted arcs..

What is the measure of angles formed by intersecting chords?

The measure of an angle formed by intersecting chords is half the sum of the measures of the intercepted arcs.

describe how the measures of a central angle and a inscribed angle are related.

The measure of the central angle is twice as much as the inscribed angle will be. it also works in reverse.

Describe how the central angle measure relates to arc length and radius.

The measure of the central angle, in radians, describes the ratio between the arc length and the radius.

center of the circle

The point at the exact __________________. All points on a circle are the same distance from the _____________.

point of tangency

The point at which a tangent line meets a curve. In a circle, the radius ending at the point of tangency is always perpendicular to the tangent line..

what do you notice about the sector that can help you find the area of a circle?

The sector covers half of the circle.

the sector

The sector of a circle is bounded by a central angle of the circle and the intercepting arc.

other then the circumference of a circle formula how might you describe the relationship between the radius and the circumference of a circle?

The size of the radius and the circumference are related. when the radius gets bigger, the circumference gets bigger by a factor of 2pi

formula for finding the area of a parallelogram.

To find the area of a parallelogram, multiply the height and the length of its base.

True or false? If two chords are congruent, they are the same distance from the center of the circle.

True

How could you show that two arcs are congruent?

Two arcs are congruent if they have the same length and belong to the same circle or two congruent circles.

When are two arcs congruent?

Two arcs of a circle are congruent if and only if their associated chords are congruent.

How do you get an estimate of the circumference of a circle from the radius?

When given the radius of a circle, multiply that number by 6 to get a rough idea of its circumference.

If a chord is bisected by a radius, is the radius perpendicular to the chord?

Yes

Do congruent chords of a circle always create congruent arcs?

Yes, congruent chords of a circle always create congruent arcs.

circumference of a circle formula

c=2pir

What can you say about the arcs of two congruent central angles

if two central angles of a circle are congruent then they form congruent arcs.

how are the areas of a sector of a circle and the circle related?

the area of a sector is a fraction of the area of the circle.

Exterior angle theorem

the exterior angle is the sum of the remote interior angles.

explair how R compares to the height of the parallelogram.

the height of the parallelogram is almost equal to R.

Arc Length

the length of an arc of a circle

what is the relationship between the measure of an inscribed angle and the measure of the arc that it intercepts.

the measure of an inscribed angle is half the measure of the arc it intercepts. it works in reverse.

What is the relationship between the radius and the tangent.

the radius is always perpendicular to the point of tangency.

Area

the space taken up by a two dimensional figure or surface. area is measured in square units.

how can you find the length of an arc using the circumference of a circle and the fraction of the circle covered by the arc?

to find arc length, multiply the cirumference of the circle by the fraction covered by the arc. (devide by fraction)

When are two chords congruent?

two chords are congruent if and only if their associated central angles are congruent.