Ex 6.3, 9

Transcript

Ex 6.4, 9 A square is inscribed in a circle of radius 𝑟. Show that the ratio of the area of the square to the area of the circle is equal to 2/𝜋≈0.637. Let’s look at it step-by-step Step 1 of 7 The Circle's Area Once again, we have a circle of radius r. Area = 𝜋r^2. Step 2 of 7 Inscribe the Square We draw a square inside the circle. Its corners touch the circle's edge. Step 3 of 7 Split with Diagonals By drawing the two diagonals of the square (which are also diameters of the circle), we split the square into 4 identical triangles.Step 4 of 7 The 〖90〗^∘ Angle Because a square's diagonals intersect at exactly 90^∘, each of the 4 pieces is a perfect right-angled triangle. Its two short sides are simply the radii of the circle. Step 5 of 7 Area of One Piece Look at one triangle. Its base is r and its height is r. " Area "=1/2×" base × height "=1/2 r^2 Step 6 of 7 Total Square Area Since there are 4 identical pieces, the total area of the square is 4×1/2 r^2=2r^2. Step 7 of 7 Calculate the Ratio Divide the Square Area by the Circle Area. The r^2 terms cancel out. Ratio =(2r^2)/(πr^2 )=2/π≈0.637