Ex 6.3, 8

Transcript

Ex 6.4, 8 An equilateral triangle is inscribed in a circle of radius 𝑟. Show that the ratio of the area of the triangle to the area of the circle is equal to (3√3)/4𝜋≈0.413. Let’s look at it step-by-step Step 1 of 8 The Circle's Area We start with a circle of radius r . The area of the entire circle is simply 𝜋r^2. Step 2 of 8 Inscribe the Triangle We draw an equilateral triangle inside the circle so that its corners touch the edge perfectly. Step 3 of 8 Split from the Center By drawing lines from the center (radii) to the three corners, we split the equilateral triangle into 3 identical isosceles triangles.Step 4 of 8 Drop a Height Let's focus on one of these slices. It has an angle of 120^∘ at the center. Dropping a straight line down splits it into two 30^∘-60^∘-90^∘ right triangles. Step 5 of 8 Find Base and Height In a 30-60-90 triangle with hypotenuse r, the short side (height) is r/2, and the long side is (r√3)/2. So the full base of our slice is r√3. Step 6 of 8 Area of One Piece Area =1/2× base × height =1/2×(r√3)×(r/2)=√3/4 r^2. Step 7 of 8 Total Triangle Area Since there are 3 identical pieces, the total area of the equilateral triangle is 3 ×√3/4 r^2=(3√3)/4 r^2. Step 8 of 8 Calculate the Ratio To find the final ratio, we divide the Triangle Area by the Circle Area. The r^2 terms cancel out perfectly! "Ratio "=(3√3/4)/π=(3√3)/4π≈0.413