Presentation on theme: "9-2 Developing Formulas for Circles and Regular Polygons Warm Up"— Presentation transcript:
1 9-2 Developing Formulas for Circles and Regular Polygons Warm Up Lesson PresentationLesson QuizHolt Geometry
2 Warm Up Find the unknown side lengths in each special right triangle. 1. a 30°-60°-90° triangle with hypotenuse 2 ft2. a 45°-45°-90° triangle with leg length 4 in.3. a 30°-60°-90° triangle with longer leg length 3m
3 ObjectivesDevelop and apply the formulas for the area and circumference of a circle.Develop and apply the formula for the area of a regular polygon.
4 Vocabulary circle center of a circle center of a regular polygon apothemcentral angle of a regular polygon
5 A circle is the locus of points in a plane that are a fixed distance from a point called the center of the circle. A circle is named by the symbol and its center. A has radius r = AB and diameter d = CD.The irrational number is defined as the ratio ofthe circumference C tothe diameter d, orSolving for C gives the formulaC = d. Also d = 2r, so C = 2r.
6 You can use the circumference of a circle to find its area You can use the circumference of a circle to find its area. Divide the circle and rearrange the pieces to make a shape that resembles a parallelogram.The base of the parallelogram is about half the circumference, or r, and the height is close to the radius r. So A r · r = r2.The more pieces you divide the circle into, the more accurate the estimate will be.
8 Example 1A: Finding Measurements of Circles Find the area of K in terms of .A = r2Area of a circle.Divide the diameter by 2 to find the radius, 3.A = (3)2A = 9 in2Simplify.
9 Example 1B: Finding Measurements of Circles Find the radius of J if the circumference is (65x + 14) m.C = 2rCircumference of a circle(65x + 14) = 2rSubstitute (65x + 14) for C.r = (32.5x + 7) mDivide both sides by 2.
10 Example 1C: Finding Measurements of Circles Find the circumference of M if the area is25 x2 ft2Step 1 Use the given area to solve for r.A = r2Area of a circle25x2 = r2Substitute 25x2 for A.25x2 = r2Divide both sides by .Take the square root of both sides.5x = r
11 Example 1C ContinuedStep 2 Use the value of r to find the circumference.C = 2rC = 2(5x)Substitute 5x for r.C = 10x ftSimplify.
12 Check It Out! Example 1Find the area of A in terms of in whichC = (4x – 6) m.A = r2Area of a circle.Divide the diameter by 2 to find the radius, 2x – 3.A = (2x – 3)2 mA = (4x2 – 12x + 9) m2Simplify.
13 Always wait until the last step to round. The key gives the best possible approximation for on your calculator.Always wait until the last step to round.Helpful Hint
14 Example 2: Cooking Application A pizza-making kit contains three circular baking stones with diameters 24 cm, 36 cm, and 48 cm. Find the area of each stone. Round to the nearest tenth.24 cm diameter36 cm diameter48 cm diameterA = (12)2A = (18)2A = (24)2≈ cm2≈ cm2≈ cm2
15 Check It Out! Example 2A drum kit contains three drums with diameters of 10 in., 12 in., and 14 in. Find the circumference of each drum.10 in. diameter in. diameter in. diameterC = dC = dC = dC = (10)C = (12)C = (14)C = 31.4 in.C = 37.7 in.C = 44.0 in.
16 The center of a regular polygon is equidistant from the vertices The center of a regular polygon is equidistant from the vertices. The apothem is the distance from the center to a side. A central angle of a regular polygon has its vertex at the center, and its sides pass through consecutive vertices. Each central angle measure of a regular n-gon is
17 Regular pentagon DEFGH has a center C, apothem BC, and central angle DCE.
18 To find the area of a regular n-gon with side length s and apothem a, divide it into n congruent isosceles triangles.area of each triangle:total area of the polygon:The perimeter is P = ns.
20 The area of a sector is a fraction of the circle containing the sector The area of a sector is a fraction of the circle containing the sector. To find the area of a sector whose central angle measures m°, multiply the area of thecircle by
22 Example 1A: Finding the Area of a Sector Find the area of each sector. Give answers in terms of and rounded to the nearest hundredth.sector HGJUse formula for area of sector.Substitute 12 for r and 131 for m.= 52.4 m2 m2Simplify.
23 Example 1B: Finding the Area of a Sector Find the area of each sector. Give answers in terms of and rounded to the nearest hundredth.sector ABCUse formula for area of sector.Substitute 5 for r and 25 for m. 1.74 ft2 5.45 ft2Simplify.
24 Check It Out! Example 1aFind the area of each sector. Give your answer in terms of and rounded to the nearest hundredth.sector ACBUse formula for area of sector.Substitute 1 for r and 90 for m.= 0.25 m2 0.79 m2Simplify.
25 Check It Out! Example 1bFind the area of each sector. Give your answer in terms of and rounded to the nearest hundredth.sector JKLUse formula for area of sector.Substitute 16 for r and 36 for m.= 25.6 in2 in2Simplify.
26 Example 2: Automobile Application A windshield wiper blade is 18 inches long. To the nearest square inch, what is the area covered by the blade as it rotates through an angle of 122°?Use formula for area of sector.r = 18 in. 345 in2Simplify.