Name_________________________________________ Date____________________ Class___________________

Developing Formulas for Circles and Regular Polygons

Find the circumference of circle S in which A 81 cm2.Step 1 Use the given area to solve for r.

A r 2 Area of a circle

81 cm2 r 2 Substitute 81 for A.

81 cm2 r 2 Divide both sides by .

9 cm r Take the square root of both sides.

Step 2 Use the value of r to find the circumference.

C 2r Circumference of a circle

C 2(9 cm) 18 cm Substitute 9 cm for r and simplify.

Find each measurement.1. the circumference of circle B 2. the area of circle R in terms of

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3. the area of circle Z in terms of 4. the circumference of circle T in terms of

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5. the circumference of circle X in 6. the radius of circle Y in which C 18 cmwhich A 49 in2

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Circumference and Area of Circles

A circle with diameter d and radius r has circumference C d or C 2r.A circle with radius r has area A r2.

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Developing Formulas for Circles and Regular Polygons continued

Find the area of a regular hexagon with side length 10 cm.Step 1 Draw a figure and find the measure of a central angle. Each central

angle measure of a regular n-gon is .

Step 2 Use the tangent ratio to find the apothem. You could also use the 30°-60°-90° Thm. in this case.

Write a tangent ratio.

Substitute the known values.

Solve for a.Step 3 Use the formula to find the area.

, P 6 10 or 60 cmA 259.8 cm2 Simplify.

Find the area of each regular polygon. Round to the nearest tenth.

Area of Regular Polygons

The area of a regular polygon with apothem a and perimeter P

is

The center is equidistant from the vertices.

The apothem is the distance from the center to a side.

A central angle has its vertex at the center. This central angle measure is

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7. 8.

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9. a regular hexagon with an apothem of 3 m 10. a regular decagon with a perimeter of 70 ft__________________________________________ __________________________________________

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Volume of Prisms and Cylinders

Find the volume of each prism.1. 2.

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Find the volume of each cylinder. Give your answers both in terms of and rounded to the nearest tenth.

3. 4.

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Volume of Prisms

Prism The volume of a prism with base area B and height h is V Bh.

Right Rectangular Prism

The volume of a right rectangular prism with length , width w, and height h is V wh.

Cube The volume of a cube with edge length s is V s3.

Volume of a Cylinder

The volume of a cylinder with base area B, radius r, and height h is

V Bh, or V r 2h.

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ReteachVolume of Prisms and Cylinders continued

The dimensions of the prism are

multiplied by . Describe the effect on the volume.

Notice that 216 8. If the dimensions are multiplied by , the volume is multiplied

by , or .

Describe the effect of each change on the volume of the given figure.

5. The dimensions are multiplied by 2. 6. The dimensions are multiplied by .

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Find the volume of each composite figure. Round to the nearest tenth.

original volume:new volume, dimensions multiplied by :

V wh V wh

    (12)(3)(6)        12, w 3, h 6

    (4)(1)(2)        4, w 1, h 2

    216 cm3         Simplify.     8 cm3           Simplify.

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7. 8.

Volume of Pyramids and Cones

Find the volume of each pyramid. Round to the nearest tenth if necessary.

Volume of a Pyramid

The volume of a pyramid with base area B and height h is

.

Volume of a Cone

The volume of a cone with base area B, radius r, and height h is

, or .

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1. 2.

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Find the volume of each cone. Give your answers both in terms of and rounded to the nearest tenth.

3. 4.

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ReteachVolume of Pyramids and Cones continued

The radius and height of the cone are

multiplied by . Describe the effect on the volume.

If the dimensions are multiplied by , then the volume is multiplied by , or .

Describe the effect of each change on the volume of the given figure.

5. The dimensions are doubled. 6. The radius and height are multiplied by .

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Find the volume of each composite figure. Round to the nearest tenth if necessary.7. 8.

original volume:new volume, dimensions multiplied by :

         r 4, h 6         r 2, h 3

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Spheres

Find each measurement. Give your answer in terms of .1. the volume of the sphere 2. the volume of the sphere

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3. the volume of the hemisphere 4. the radius of a sphere with volume

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5. the surface area of the sphere 6. the surface area of the sphere

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Volume and Surface Area of a Sphere

Volume The volume of a sphere with radius r is

.

Surface Area The surface area of a sphere with radius r isS 4r2.

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ReteachSpheres continued

The radius of the sphere is multiplied by . Describe the effect on the surface area.

Notice that 1024 64. If the dimensions are multiplied by ,

the surface area is multiplied by , or .

Describe the effect of each change on the given measurement of the figure.

7. surface area 8. volume

The radius is multiplied by 4. The dimensions are multiplied by .

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Find the surface area and volume of each composite figure. Round to the nearest tenth.

9. Hint: To find the surface area, add the 10. Hint: To find the volume, subtract the lateral area of the cylinder, the area of volume of the hemisphere fromone base, and the surface area of the the volume of the cylinder.hemisphere.

original surface area:new surface area, radius multiplied by :

S 4r 2 S 4r

2

    4(16)2      r 16 4(4)2      r 4

    1024 m2  Simplify 64 m2    Simplify.

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Lines That Intersect Circles

Identify each line or segment that intersects each circle.1. 2.

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Find the length of each radius. Identify the point of tangency and write the equation of the tangent line at that point.

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Holt McDougal Analytic Geometry

Lines and Segments That Intersect Circles

• A chord is a segment whose endpoints lie on a circle.

• A secant is a line that intersects a circle at two points.

• A tangent is a line in the same plane as a circle that intersects the circle at exactly one point, called the point of tangency.

• Radii and diameters also intersect circles.

Tangent Circles

Two coplanar circles that intersect at exactly one point are called tangent circles.

E is a point of tangency.

and are chords.

is a secant.

is a tangent.

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3. 4.

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Holt McDougal Analytic Geometry

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ReteachLines That Intersect Circles continued

In the figure above, EF 2y and EG y 8. Find EF.EF EG 2 segs. tangent to from same ext. pt. segs. .

2y  y 8 Substitute 2y for EF and y 8 for EG.

y 8 Subtract y from each side.

EF 2(8) EF 2y; substitute 8 for y.

16 Simplify.

The segments in each figure are tangent to the circle. Find each length.

5. BC 6. LM

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7. RS 8. JK

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Holt McDougal Analytic Geometry

Theorem Hypothesis Conclusion

If two segments are tangent to a circle from the same external point, then the segments are congruent.

and are tangent to .

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Holt McDougal Analytic Geometry

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Arcs and Chords

Find each measure.

1. ________________________ 3. ________________________

2. ________________________ 4. ________________________

5. ________________________

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Holt McDougal Analytic Geometry

Arcs and Their Measure• A central angle is an angle whose vertex is the center of a circle.• An arc is an unbroken part of a circle consisting of two points on a circle and all the points

on the circle between them.

• If the endpoints of an arc lie on a diameter, the arc is a semicircle and its measure is 180°.

Arc Addition Postulate

The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.

is a major arc.

360° mABC 360° 93° 267°

ABC is a central angle.

is a minor arc

mABC 93°.

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6. ________________________

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Holt McDougal Analytic Geometry

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ReteachArcs and Chords continued

Congruent arcs are arcs that have the same measure.

In a circle, if a radius or diameter is perpendicular to a chord, then it bisects the chord and its arc.

Find each measure.

7. . Find . 8. HLG KLJ. Find GH.

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Find each length to the nearest tenth.9. NP 10. EF

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Holt McDougal Analytic Geometry

Congruent Arcs, Chords, and Central Angles

If mBEA mCED,

then If , then If , then

mBEA mCED.

Congruent central angles have congruent chords.

Congruent chords have congruent arcs.

Congruent arcs have congruent central angles.

Since

bisects and

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ReteachSector Area and Arc Length

Find the area of each sector. Give your answer in terms of and rounded to the nearest hundredth.

1. sector CDE 2. sector QRS

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Holt McDougal Analytic Geometry

Sector of a Circle

A sector of a circle is a region bounded by two radii of the circle and their intercepted arc.The area of a sector of a circle is given by the

formula .

Segment of a CircleA segment of a circle is a region bounded by an arc and its chord.

sector ABC

segment ABC

area of segment ABC

area of sector ABC

area of ABC

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Find the area of each segment to the nearest hundredth.3. 4.

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ReteachSector Area and Arc Length continued

Find the arc length of

Formula for arc length

Substitute 9 cm for r and 84° for m°.

Simplify.

13.19 cm Round to the nearest hundredth.

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Holt McDougal Analytic Geometry

Arc Length

Arc length is the distance along an arc measured in linear units.

The arc length of a circle is given by the formula

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Find each arc length. Give your answer in terms of and rounded to the nearest hundredth.

5. 6.

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7. 8.

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Holt McDougal Analytic Geometry

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Inscribed Angles

Find each measure.

1. mLMP and 2. mGFJ and

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Find each value.3. x 4. mFJH

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Holt McDougal Analytic Geometry

Inscribed Angle Theorem

The measure of an inscribed angle is half the measure of its intercepted arc. is an

intercepted arc.

ABC is an inscribed angle.

Inscribed Angles

If inscribed angles of a circle intercept the same arc, then the angles are congruent.ABC and ADC intercept

so ABC ADC.

An inscribed angle subtends a semicircle if and only if the angle is a right angle.

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ReteachInscribed Angles continued

Find mG.Step 1 Find the value of z. mE mG 180° EFGH is inscribed in a circle. 4z 3z 5 180 Substitute the given values.

7z 175 Simplify.z 25 Divide both sides by 7.

Step 2 Find the measure of G.mG 3z 5

3(25) 5 80° Substitute 25 for z.

Find the angle measures of each quadrilateral.5. RSTV 6. ABCD

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7. JKLM 8. MNPQ

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Holt McDougal Analytic Geometry

Inscribed Angle Theorem

If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.

A and C are supplementary.B and D are supplementary.

ABCD is inscribed in

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Holt McDougal Analytic Geometry