Math MagazineSpecial Double Issue Edition

Circles

Area of Polygons and Circles

The Royal Wedding

Careers Using Circles and Area

Table of Contents

Page3 to 1516 to 24 25 to 30 

ContentsCircles and CareersArea of Polygons and Circles and CareersThe Royal Wedding, Real-Life Applications and Quiz withAnswer Key   

Circles

In this section about circles you will learn about circles and their properties.  You will also learn about arcs on circles, angles and segments related to circles, and how to graph circles.

Tangents to Circles

A circle is a set of points that are all equidistant from the center of the circle.  The distance from any of these points to the center is the radius, circles are congruent if they have the same radius.  The distance across a circle, through the center, is the diameter.  A segment whose endpoints are on the circle is a chord while a segment that intersects a circle in two points is secant.  A line that intersects the circle at one point is a tangent.  The point that the line intersects the circle is the point of tangency. Circles that lie on the same plane and intersect at one point are tangent circles while those who share a center are concentric.  If a line is tangent to coplanar circles it is called a common tangent.  Chiropractors who treat patients with many problems including spinal injuries use circles thier work and use their scientific knowledge.

Theorems on Tangents to Circles

Theorem 10.1 states that if a line is tangent to a circle, then it is prependicular to the radius drawn to the point of tangency. Theorem 10.2 states the in a plane, if a line is perpendicular to a radius of a cirlce at its endpoint on the circle, then teh line is tangent to the circle. Theorem 10.3 states that if two segments from teh same exterior point are tangent to a cirlce, then they are congruent.

Arcs and Chords

An angle whose vertex is the center of a circle is a central angle.  If the central angles measures less than 180 it creates a minor arc on the circle, if it measures more than 180 it creates a major arc.  If the endpoints of an angle are on the diameter then the arc is a semicircle.  The measures of the arcs are equal to the measures of the central angles that create them.  Two arcs can be congruent if they have the same measure and are the same circle or congruent circles.  There is one postulate dealing with arcs, the Arc Addition Postulate.  This states that the measure of an arc formed by two adjacent acrs is the sum of the measures of the two arcs.  Astronomers use cirlces in their mapping of the universes along with creativity.

Theorems on Arcs and Chords

Theorem 10.4 states the in the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.  Theorem 10.5 states that if a diameter of a cirlce is perpendicualr to a chord, then the diameter bisects the chord and its arc. Theorem 10.6 states that if one chord is a perpendicular bisector of another chord, then the first chord is a diameter. Theorem 10.7 states that in the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center.

Inscribed Angles

Inscribed angles are angles whose vertex is on the circle and whose sides contain chords of the circle.  The arc that is created by the angle is callled the intercepted arc.  Polygons whose vertices all lie on a cirlce are inscribed in the circle while the circle is circumscribed about the polygon.  Physicists study the matter that makes up the universe and natural forces such as gravity, they use cirlces in their studies on a regular basis.  Physicists also use scince and technology on regular basis.

Theorems on Inscribed Angles

Theorem 10.8 states that if an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc.Theorem 10.9 states that if two inscribed angles of a circle intercept the same arc, then the angles are congruent.  Theorem 10.10 states that if a right triangles is inscribed in a circle, then the hypotenuse is a diameter of the circle.  Conversely, if one side of an inscribed triangles is a diameter of the circle, then the trianlges is a right triangle and the angles opposite the diameter is the right angle.Theorem 10.11 states that a quadrilateral can inscribed in a circle if and only if its opposite angles are supplementary.

Other Angle Relationships in Circles

Two lines can intersect a circle in three different places.  They can intersect on the circle, inside the circle, or outside the circle.  Boilermakers use circles in making boilers, vats, and other large tanks and innovation to create new methods of building vats, ect. 

Theorems on Other Relationships

Theorem 10.12 states that if a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc.Theorem 10.13 states that if two intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.Theorem 10.14 states that if a tangent and a secant, two tangents, or two secants intersect in the exterior of a cirlce, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.

Segment Lengths in Circles

Tangent segments are tangents that tangent to a cirlce at their endpoint.  Secant segments are the segments of a secant that go from a given point to the opposite side of the circle.  The external secant segment is the part that is on the exterior of the circle.  Drafters use circles in their technical drawings of spacecraft to office buildings they also use pre-enginneering teachings and creativity in their work.

Theorems on Segment Lengths

Theorem 10.15 states that if two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord. Theorem 10.16 states that if two secant segments share the same endpoint outside a circle, then the product of the length of one secant segment and the length of its external segment equals the porduct of the length of its external segment. Theorem 10.17 states that if a secant segment and a tangent segment share an endpoint outside a circle, then the product of the length of the secant segment and the length of its external segment equals the square of the length of the tangent segment.

Equations of Circles

When writing the equation of a circle the radius will be represented by r, the center by (h,k), and any given point on the circle by (x,y).  You can then put this into the distance formula and to find the standard equation of a circle you can square both sides.  This gives you (x-h)2+(y-k)2=r2.  If the center of the circle is the orgin then the equation is x2+y2=r2.  Machinists, who make specialized parts of advanced techonology, use circles in their product frequently with technology.

Locus

Locus in a plane is the points that satisfy the given condition or conditions.  Locus is derived form the Latin word for location.  The plural for locus is loci.  Compter software engineers use circles, and loci, very often when designing equations for software with math and science in unison with innovation.

Area of Polygons and Circles

In this section about area of polygons and circles you will learn about angle measures and areas of polygons.  You will also learn how to compare perimeters and areas of similar figures.  You will lastly learn how to the find the circumference and area of a circle along with other measures.

Angle Measures in Polygons, Theorems

To understand angles measures in polygons there are two theorems and corollaries.  Theorem 11.1 states that the sum of the measures of the interior angles of a convex n-gon is (n-2)x180.  The corollary to this theorem states that if the measure of each interior angle of a regular n-gon is (1/n)x(n-2)x180, or (n-2)x180/n.  Theorem 11.2 states that the sum of the measures of the exterior angles of a convex polyogn, one at each vertex, is 360.  The corollary to this theorem states that the measure of each exterior angle of a regular n-gon is (1/n)x360, or 360/n.  Landscape architects use polygons and their angle measures to create appealing outdoor areas for clients with creativity.

Areas of Regular Polygons

The center of a polygon and radius are the same as those of the polygon's circumscribed circle.  The apothem of a polygon is the distance from the center to any side of the polygon.  This is also the height of a triangle made with the radii to the consecutive vertices.  The central angle of a polygon is an angle that is formed with the center and two consecutive radii.  In a regular polygon a central angle measurement can be found by dividing 360 by the number of sides.  Real estate agents use area to find the area in their houses and buildings while using the 21st century skill of math. 

Areas of Regular Polygons Theorems

Theorem 11.3 states that the area of an equilateral triangle is one fourth the square of the length of the side times the square root of three. Theorem 11.4 states that the area of a regular n-gon with side length s is half the product of hte apothem a and the perimeter P.  

Perimeters and Areas of Similar Figures Theorems

To understand the realtionship between perimeter and area of similar figures you must understand this theorem. Theorem 11.5 states that if two polygons are similar with the lengths of corresponding sides in the ratio of a:b, then the ratio of their areas is a2:b2. Agricultural workers use area and perimeter frequently in their farming and planning along with science and innovation.

Circumference and Arc Length

The circumference of a circle is the distance around the circle.  The ratio of the circumfernce to the diameter is pi.  Arc length is a given portion of the circumference of a circle.  The lengths of a semicircle and a 90 degree arc are half the circumference and one fourth of the circumference, respectively.  Police and detectives use cirlces in their mapping of areas for many different reasons along with creativity to solve cases.

Circumference and Arc Length Theorems

Theorem 11.6 states the circumference C of a circle is C=(pi)d or C=2(pi)r, where d is the diameter of the circle and r  is the radius of the circle. Arc Length Corollary states that in a circle, the ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to 360 degrees. 

Areas of Circles and Sectors Theorems

Theorem 11.7 states that the area of a circle is pi times the square of the radius. A sector is a region of a circle bound by two radii and their intercepted arc. Theorem 11.8 states that the ratio of the area of a sector of a circle to the area of the circle is equal to the area of the measure of the intercepted arc to 360 degrees. Aerospace engineers use circles (and math, science, and pre-engineering) in their work designing, contructing, and testing different types of aircraft while using creativity, technology, and innovation. 

Geometric Probability

Probability is the likeness of an event to occur represented by a number from zero to one.  Geometric probability is a version of probability that involves length and area.  Atmospheric scientists, meteorologists, use geometric probability in their predicition of where events will occur along with technology.

Royal Wedding Quiz

This is the Duke and Duchess of Cambridge's wedding cake.  If Catherine slices a piece from the center of the top layer with an anlge of 45 degrees what is the measure of the arc that is the outside of the piece of cake?Supposed that the area of the piece of cake is 14 and one eighth of a square inch, what is the area of the top layer? 

Answers

45 degrees113 square inches

From the center of the Duchess of Cambridge's bouquet to the outer edge there are only type of flowers ina certain area.  The whole area of the bouquet is 7 inches and the arc length of the given area is 80 degrees.  What is the area of this sector?

Answer

1.5 square inches

This is the a section of the floor in Westminster Abbey where the marriage ceremony took place.  What is the geometric probability that the ceiling will collapse above the smaller square if it has side lengths of 14 feet inside the larger square with side lengths of 20 feet?

Answer

49%

 

Identify where the central angle, radius, and apothem of this window in Westminster Abbey.

 

Answer

 

 

These photographs show the soon to be Duchess of Cambridge riding in the Queen's Rolls-Royce Phantom VI and Prince William through similar quadrilateral windows.  If the ratio of sides is 2:3, what is the ratio of areas of the windows?

 

Answer

4:9

Created by Carly Cannoy