Ex 6.3, 10

Transcript

Ex 6.4, 10 A hexagon is inscribed in a circle of radius 𝑟. Show that the ratio of the area of the hexagon to the area of the circle is equal to (3√3)/2𝜋≈0.827. Can you see why the answer is exactly twice the answer to Question 8? Let’s look at it step-by-step Step 1 of 9 The Circle's Area Circle of radius r . Area = 𝜋r^2.Step 2 of 9 Inscribe the Hexagon We draw a regular hexagon inside the circle. Step 3 of 9 Split from the Center By drawing radii to all 6 vertices, we split the hexagon into 6 identical triangles. Because 360^∘/6=60^∘, these are all perfect equilateral triangles with side length r! Step 4 of 9 Drop a Height Let's focus on one equilateral triangle at the bottom. We drop a perpendicular line down the middle to find its height. This splits its base r perfectly in half Step 5 of 9 Find the Height Using the Pythagorean theorem (or 30-60-90 rules), if the hypotenuse is r and the base is r/2, the height is (r√3)/2Step 6 of 9 Area of One Piece Area " =1/2× base × height =1/2×r×((r√3)/2)=√3/4 r^2.) Step 7 of 9 Total Hexagon Area Since there are 6 identical pieces, the total area is 6×√3/4 r^2=(6√3)/4 r^2=(3√3)/2 r^2. Step 8 of 9 Calculate the Ratio Divide the Hexagon Area by the Circle Area. The r^2 terms cancel out. Ratio =(3√3/2)/π=(3√3)/2π≈0.827 Step 9 of 9 Why exactly TWICE Question 8? Look closely: The hexagon is made of 6 identical equilateral triangles. The triangle from Question 8 (highlighted) perfectly covers exactly 3 of them! Since 6 is exactly double 3 , the hexagon's area is exactly double the triangle's area.