FW 3.4: More Circle Practice Learning Target: I will find the area of circle by connecting to the area of a parallelogram. HW: Complete Inv. 3.4 p. 9, Title of Picture puzzle and CORRECT with the KEY. Warm Up: What is the constant of proportionality of the line represented by the equation What is the y-intercept of the line represented by the equation 2 3 y = x - 4 2 3 y = x - 4 Solve the equations. 2. 10b – 45 = 4(b -15) 3. 4(w - 6) = 3(w + 1)

Warm-Up Solve the equations. 2. 10b – 45 = 4(b -15) + 45 + 45 10b = 4b + 15 - 4b -4b 6b = - 15 ÷6 ÷ 6 b = -15/6 = -5/2 = -2.5 3. 4(w - 6) = 3(w + 1) 4w – 24 = 3w + 3

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Circumference = dπ = 12π = 12 x 3.14 = 37.7 inches Area = πr² = 36π = 113.1 square inches / 113.1 in.² Answers involving pi were calculated using the π key on a calculator. If students use 3.14 for pi, their answers may vary slightly.

Height: The circle sectors have radius 6 inches, and those radii are close to the heights of the triangles that make up the parallelogram. Base: The arcs of the circle sectors together comprise the circumference of the circle. They approximate the bases of the triangles that make up the parallelogram—half on the top and half on the bottom of the parallelogram. Base and Height. In the approximation by circle sectors this means (radius)(half circumference) = (6 inches)(6π inches) = 36π square inches. They are the same since the circle has simply been cut into sectors that are reassembled to approximate the parallelogram.

The two formulas are equivalent The two formulas are equivalent. The approximation of parallelogram area leads to the same calculations as area of a circle suggested by the radius square exploration. The two formulas will work in the same way for circles of different radii. Since the radius of a circle is half its diameter, r=12d, regardless of the value of the radius. The arcs would be closer to straight-line segments making up bases of the triangles covering the parallelogram, and the radii would be closer to heights of those triangles.

Optimal area for fixed perimeter occurs with the circle. D. Suppose that you have 12 meters of fencing and want to make a pen for your pet dog. Which shape, a square or a circle, would give more area? Explain. Hint: Remember our polygonal prisms and rice! Square: the side lengths = 3 m. each area = 9m². Circle: circumferencer: 12/π ≈ 3.8 meters, radius ≈1.9meters, and area ≈11.3 m². Optimal area for fixed perimeter occurs with the circle.  E. A rectangular lawn has a perimeter of 36 meters and a circular exercise run has a circumference of 36 meters. Which shape will give Rico’s dog more area to run? Explain. Rectangle (Square): side lengths = 9 meters each , area = 81m². Circle: 𝒄𝒊𝒓𝒄𝒖𝒎𝒇𝒆𝒓𝒆𝒏𝒄𝒆 𝒑𝒊 = 𝟑𝟔 𝟑.𝟏𝟒 ≈ 11.46 diameter d=2r 𝟏𝟏.𝟒𝟔 𝟐 ≈ 5.73 πr² ≈ 5.73 x 5.73 x 3.14 = 103.13m² Optimal area for fixed perimeter occurs with the circle.

FW 3.4: Circumference Did I reach my Learning Target? I will find the area of circle by connecting to the area of a parallelogram. HW: Complete Inv. 3.4 p. 9, Title of Picture puzzle and CORRECT with the KEY.