Final answer:
The total non-shaded area consists of the sectors minus the area of the equilateral triangle. Calculate the triangle area with (√3/4) * side² and sector area with (60/360) * π * r², and then subtract the triangle area from the total area of the sectors.
Step-by-step explanation:
To calculate the total area of the non-shaded region of the diagram that includes arc AB, arc AC, and arc BC, we need to consider an equilateral triangle and the segments of the circles that form when circles are drawn with centers at each vertex of the triangle, and radius the same as the side of the triangle.
The first step is to find the area of the equilateral triangle ABC. The formula for the area of an equilateral triangle is (√3/4) * side². With each side being '5 cm', the area is:
Area of triangle ABC = (√3/4) * 5² = (√3/4) * 25 = (25√3/4) cm²
To find the area of each sector (which is a slice of a circle), we can use the fact that the angle at the center of the sector is 60 degrees since it's part of the equilateral triangle. The area of a sector is given by (θ/360) * π * r², where 'θ' is the central angle in degrees and 'r' is the radius. Therefore, for one sector:
Area of sector = (60/360) * π * 5² = (π/6) * 25 cm²
Since there are three identical sectors, we multiply the area of one sector by 3 to get the total area of all sectors. Finally, subtract the area of the triangle from the total area of the sectors to get the total non-shaded area.