Geometry Questions

Find the volume V of the described solid S The base of S is the region enclosed by the parabola y 6 x and the x axis Cros perpendicular to the x axis are squares

3 You work for a company that makes toy marbles The radius of the most popular marble is 0 3 inches The volume of a marble is given by the formula V where 3 14 ch out of every d a Rounded to the nearest tenth of a cubic inch what is the volume of glass needed to make one of the most popular marbles b What volume of glass is needed to make 105 marbles

Solve problems involving the of a circle central angle circumference Solve problems involving Words to Know Write the letter of the definition next to the matching word as you work through the lesson You may use the glossary to help you arc length radian Define measure A a standard unit of measure for angles the measure of a central angle that subtends an arc that is equal in length to the radius of the circle B an angle whose vertex is at the center of a circle and whose sides are radii of that circle BOK C a portion of the circumference of a circle MAGNE AMOR ARRICANE D the distance around a circle

Circumference of a Circle The circumference of any circle or the distance around the circle can be determined using either of these formulas 2 m r radius C md d Applying Circumference The London Eye is a Ferris wheel that at its highest point stands 135 meters tall The diameter of the London Eye is 120 meters A person rides the wheel two full revolutions How far has the person traveled Calculate the circumference Use the button on your calculator C nd C C 377 m Find the distance in two full revolutions 377 2 F O m

Relating Radlans and Degrees The exact number of radians in one half of a circle is w A semicircle has half the circumference C In terms of radians the measure of the central angle is equal to the arc length divided by the radius 8 A semicircle has 180 E 180 T rad There are about 3 14 radians in a semicircle 0 14r

C 2mr or C nd A radian is a central same length as the radius In degrees s Arc length a portion of the circumference of a circle can be found using In radians the distance around the circle is found using where the intercepted arc is the 1 radian r ww

Arc Length Arc length s is a c 2 C C 10m Use 3 14 as an approximation for this is of the whole circle 4 For any arc length s of the circumference of a circle S SA S A circle has 360 This arc has a central angle of 90 That means that 8 360 2 r 31 4 A where 8 is the central angle and r is the radius of the circle O 90 5 S B

Finding Radian Measure Find the number of radians in one fourth of a circle half of a semicircle or a fourth of 1 2 2 4 1 two thirds of a circle 2 3 Examining Arc Length When r 1 S 0 27 The units for the prel m or If the radius of a circle is equal to 1 0 0 2n S 277 1 S 2 1 H N 2TT 2 2 3 1 3 rad rad Tradians 2 0 0 K 360 S 0 L

Finding Arc Length If e 15 what is the length of AB 0 S S S S 360 angle whose 24 7 12 147 T 2 r 2 Radian Measure One radian is the measure of a central is the t As B 70 A 0 1 rad S r

If the central angle has a measure of 1 radian it will subtend an arc that has the same length as its radius regardless of the size of the circle 0 Or 0 A 2 B s 0r When finding arc length use s 8r O D CE R 0 A R Finding Arc Length with a Central Angle Measured in Radians In circle O 0 2 6 radians What is the length of minor arc DE S S B S 31 2 cm

Circumference and Area of a Circle The circumference of a circle is the distance arc length s is the distance C 2 r 2 TT 31 4 units 3 14 is an approximation for 7 B MAB C 360 107 60 360 31 4 52 units the entire circle while two points on the circle 60 5 The arc length is as long as the entire circumference because the arc 6 represents of the circle N S B

The area of a circle is determined by the radius of the circle with the formula The area of a creating the sector Area of sector Area of sector dimens 0 360 is determined by the size of the central angle 0 when the central angle is measured in when the central angle is measured in

What is the area of the shaded sector created by central angle DOE 0 2 T 0 277 7 r 3 t 4 8 1 4 257 D 0 37 4 75 5 r 3 15 E units

Area of a Circle The area of a circle is found by multiplying the times Area is the amount of space inside of a figure All circles are similar because any full circle is measured to be 360 degrees or 2 pi in radian measure The area is only affected by the changes in the radius KO 6 feet Solving Problems Involving Area A fountain with a circular base has a radius of 6 feet and is surrounded by a circular path that has a width of 2 feet What is the total area occupied by the path alone 2 feet r Fountain Find the area of the fountain find the area of the fountain and the path and then subtract them from one another Area of the fountain A A r Area of the path A 6 R of the radius Area of the fountain and path r 36 ft 9 64 ft 367

TULE DIUS 10 the area of the sector central DOS MONETANSARE Words to Know Write the letter of the definition next to the matching word as you work through the lesson You may use the glossary to help you CHERRIMON PERSO VARIND degree seco Minthem OPT S radian DASS AMOUR A Do for BOSTA MRISH SKECH SONG SONGS HAGE angle PHARETLY SATRZAS ELETT Pinces ARMELEG enrised Recor CARLMENTEER INSTAT me ke Analyze the sector of a circle D an angle whose vertex is at the center of a circle whose and sides are radii of that circle CHOLERNISS MARKE a circle veltestant A a standard unit of measure for angles the measure of a central angle that subtends an arc that is equal in length to the radius of the circle of B the region of a circle bounded by two radil and their intercepted arc C a unit of measure used in measuring angles and arcs there are 360 degrees in a circle P HONGE Sicht El de marque HAMAINEE Es Polaget TRAINE PLANT S Pes WANA AME AMPAST SA

When the measure of the sector is found by multiplying these two factors The ratio of the central angle over The area of the Area of a sector Ag nr 0 7 7 7 8 2 1 1 70 4 units is given in radians the area of a 2 2 Z Y 8

Sectors of Circles A sector of a circle is the region of a circle bounded by two intercepted Calculating Area of a Sector The area of a How large the central angle is in relation to the measure of the entire circle determines how big the sector is in relation to the entire circle The A The area of the B T C x Area of a sector r 360 of a circle is found by multiplying these two factors of the central angle over 360 degrees circle B and their 135 16

When a arc or different congruent arcs the central angle an angle will be supplementary angles mAOB m2ACB 180 1 angle and circumscribed angle intercept the Solving for Unknown Measures What is the measure of angle D The arc intercepted by an inscribed angle is twice the measure of that angle TR B 43 x 2 86 The measure of the central angle is the same as the arc it intercepts m BOC 86 mz The circumscribed angle is supplementary to the central angle 180 m m AOB mZWYX ST W 43 0 X B CO www 180 B 86 D O

43 9 km h 12 7 km km Area 81 9 km A 12 4 km C 10 4 km B 11 9 km D 9 1 km

For central angles and inscribed angles that intercept the same arc W W M 1 the measure of the inscribed angle is the measure of the W angle when the intercepted arc is a semicircle a right triangle is created when the chords of the inscribed angle are congruent a For central angles and circumscribed angles that intercept the same arc created the central angle and the circumscribed angle are the angles form a kite angles is

38 1 4 km A 6 km C 8 6 km 2 6 km 2 km 2 B 8 6 km D 6 8 km

and tangents to a Compare lengths of of a circle Compare of a circle Words to Know Fill in this table as you work through the lesson You may also use the glossary to help you an angle whose vertex is on a circle and whose sides are chords a segment with both endpoints on a circle an angle whose vertex is at the center of a circle and whose sides are radii of that circle a segment of a tangent that has an endpoint at the point of tangency an angle whose vertex is outside of a circle and whose sides are tangents to that circle two angles whose measures have a sum of 180

An inscribed angle has two Intersecting chords with the vertex and the endpoints on the circle The sides are not necessarily A circumscribed angle is created by two intersecting segments The vertex is outside of the circle The sides are congruent 0

When an inscribed angle intercepts a inscribed angle create a triangle The central angle is a straight angle so it has a measure of 180 The inscribed angle intercepts the same arc as the central angle so the measure of the inscribed angle is of 180 or 90 2 When the sides of the inscribed angle are congruent they create a 45 45 90 triangle the central angle and B Quadrilaterals Created by Central and Inscribed Angles GENERAL CASE What polygon is created when the inscribed angle does not intercept a diameter of the circle B Draw tick marks to show that the radil are congruent

40 10 m 8 m A 5 m C 12 m 6 m B 24 m D 14 8 m

What polygon is created when the Inscribed angle does not intercept a diameter of the circle N If the chords are congruent the quadrilateral formed is a kite The diagonals are The angles formed by the two non congruent sides are The diagonal that runs from the vertices of the two non congruent sides will be bisected 1

4 7 km km A 5 2 km C 3 km 4 8 km Area 14 7 km B 4 2 km D 5 7 km

Relax How are the measures of inscribed angles and central angles related when they intercept the same arc or congruent arcs The Intercepted arc has the same measure as the central angle N The measure of the inscribed the intercepted arc will be half the measure of When a mzACB 1 of the 50 An inscribed angle that intercepts the same arc as a central angle will be of the measure of the central angle Determining Measures of Inscribed and Central Angles mAB mZAOB O congruent arcs the measure of the central angle will be angle 2 m AJB 0 angle and inscribed angle intercept the same arc or mZQOM 2 mZPRS O B P the measure

Relating Central and Circumscribed Angles UNDERSTANDING THE RELATIONSHIP How are the measures of circumscribed angles and central angles related when they intercept the same arc or different congruent arcs The intercepted arc has the same measure as the central angle THE The entire arc of a circle is 360 1 2 0 250 The measure of the circumscribed angle is half the of the intercepted arcs mADB mAB 110 70 D 0 110 B C

If the perimeter of kite NMOL is 42 units what is the length of side NL What is the length of diagonal NO 42 9 This is the combined length of NM and NL so divide by 2 NL 12 The point of tangency between a radius and a tangent segment is a right angle We can use the Pythagorean theorem to find NO NO 12 9 144 N 12 Angit M 9 O

What polygon is created by central and circumscribed angles that intercept the same arc The quadrilateral is a The radii and the tangents create right angles 6 The diagonals are perpendicular One diagonal is bisected H N M

Find the area of each 39 2 8 mi 3 3 mi A 12 24 mi C 18 48 mi 6 4 mi 3 3 mi B 9 24 mi D 4 6 mi

D PROVING THEY ARE Given Angle ACB is a circumscribed angle that intercepts the same arc as central angle AOB Prove m2ACB m2AOB 180 0 SUPPLEMENTARY B L m20AC m 0BC Tangents are perpendicular to radii at the point of tangency O m20AB mZ0BC m2AOB m2ACB The sum of the interior measures of a quadrilateral is 360 Subtract the measures of the two right angles m2ACB m2AOB 360 90 90 m2ACB m2AOB O

Find the peremiter of each 37 10 ft 6 1 ft 7 ft A 26 ft C 23 1 ft 9 ft B 25 1 ft D 26 ft

41 3 6 km 3 4 km 3 km A 17 1 km C 34 2 km 8 km 4 km B 8 6 km D 20 8 km

2 10 3 ft 9 2 ft 9 2 ft A 94 76 ft C 88 06 ft 10 3 ft B 189 52 ft D 47 4 ft

Central Inscribed and Circumscribed Angles Central angles circumscribed angles and inscribed angles are created tangent segments W W and outside of a circle by A chord is a segment that has two on the circle A tangent segment has an endpoint at the point of tangency that are Central angles are created by two radii They have a vertex at the of the circle The endpoints are on the circle and the sides are congruent radii chords and

11 4 points Mr Longo and Ms Johnson are celebrating the end of the year by getting ice cream Mr Long ordered 1 scoop of ice cream with a radius of 5 cm and Ms Johnson ordered 2 scoops both with a diame of 6 cm Given that each of their ice cream cones have a volume of 121 cm who has the most ice cream Explain your reasoning

Identify the and radius of a circle center Examine equations of circle hypotenuse radius given in standard form or general form Write the of a circle Words to Know Write the letter of the definition next to the matching word as you work through the lesson You may use the glossary to help you Determine if a given lles on a circle A the set of all points in a plane that are a given distance away from a given point called the center B a segment that extends from the center of a circle to any point on the circle C the fixed point that is equidistant from all points on a circle D the side of a right triangle that is opposite the right angle and is always the longest side of the triangle

The form of an equation of a circle is x h y k r The point h k is the center and is the The general form of an equation of a circle is x y Cx Dy E 0 1 Complete the form to convert from general form to standard

on the Circle the Equation Given the center and a Point Find the equation of the circle with a center at 1 5 that passes through the point 3 8 1 Use the d x x y y Let x y 1 5 and xz 3 8 formula to determine the radius T 13 3 1 8 5 4 8 5 V449 Substitute the known values into the standard form x h y k r x 1 y 5 13 x 1 y 5

10 3 points A water cup in the shape of a cone has a diameter of 8 inches and a height cup is filled with water to half of its height determine the volume of water in the cup to the nearest hundredth of an inch Hint The height and radius are proportional in a cone so if the height is cut in half so is the radius

4 The following represent three different shaped cups I rectangular prism with a square base with a side length of 3 in and a height of 6 in II cylinder with a diameter of 4 in and a height of 3 in III cone with a diameter of 6 in and a height of 5 in Which of the following represents the cups in order of greatest volume to least volume A I II III B I III II C II III I D II I III

Use the scenario below to answer 6 and 7 A rectangular prism fish tank has a base with a length of 40 inches a width of 35 inches and a height of 12 inches 6 If the tank is filled to 75 capacity what is the volume in the tank A 360 cubic inches B 480 cubic inches C 12 600 cubic inches D 16 800 cubic inches 7 Water is removed from the rectangular prism tank such that the water is at a height of 2 inches What is the volume in the tank now A 80 cubic inches B 840 cubic inches C 2 800 cubic inches D 16 800 cubic inches

Identifying the Center and Radius Given an Equation in General Form Identify the center and radius of a circle whose equation is x y 10x 14y 58 0 Complete the square for the x and y terms Group together the x terms x 10x y 14y Determine what value needs to be added to both sides of the equation to complete the square y 14y x 10x 58 25 49 16 Factor the polynomials Identify the center and radius x y ap 1 71 214 X 2 2 19 2 25 14 24 y 49

The General Form of an Equation of a Circle The To convert an equation in general form to standard form square form of an equation of a circle is x y Cx Dy E 0 Complete the square for the x and y terms Group together the x terms x 4x y Determine what value needs to be added to both sides of the equation to complete the square x 4x 4 y 8y 16 10 4 16 10 Factor the polynomials x y 4x 8y 10 0 1 T 10 h k 2 10 2 41 the 2 4 2 2 20 y

Determining Whether a Point Lies on a Circle The point 4 0 lies on a circle that is centered at the origin Does the point 2 12 also lie on the circle d x x y y All radii of a circle have the same length r 4 R Is 2 12 4 units from the center x y 0 x2 y 2 4 2 0 12 0 16 Yes the point does lie on the circle 5 5 4 3 2 1 4 2 0 0 1 2 7 5 y 1 2 3 31 5

5 A circular town with a maximum diameter of 4 miles The shaded region represents the water The land is in the shape of a rectangle There are currently 2 052 people living in the town below What is the population density in person per square mile of land A 128 3 B 163 3 C 213 2 D 698 0 4 miles OURIO 3 5 miles s dalwo003 A 2 75 miles

10 29anoq o bobna squ 12 2 points On Saturday July 1st Yellowstone National Park had 25 000 visitors in the park at 1 00 PM The total area of the park is 8 983 13 km and the total area of water is 2 714 13 km Determine the density of people on land in Yellowstone National Park on Saturday July 1st at 1 00 PM to the nearest visitor

The standard form of an equation of a circle centered at the x y r where r is the radius The hypotenuse of the triangle is a length is r Find the lengths of the legs The 1 PS y Substitute into the Pythagorean theorem y r where r is the Identifying the Radius and Center from an Equation h k 1 0 x 3 y 4 25 4 of the circle so its Q 0 0 and h k is the center of the circle form of an equation of a circle is x h y k r 2 x y 7 19 h is the value subtracted from x is the value Is T 19 h k 0 P x y S x 0 from