Problem
The touching point is one radius from each center, so center-to-center = radius + radius = diameter: each side is one diameter.
Two radii laid end to end across the touch point make exactly one diameter.
3.G.A.1Draw A DiagramEach side of the triangle -- the segment joining the centers of two touching, equal circles -- is exactly one diameter long.
The side is the distance from one center to the other, and where the two equal circles touch, that distance is one radius plus one radius -- the same length as a diameter.
The point where the circles touch sits on the straight line between the two centers and lies on both circles, so the center-to-center segment breaks into a radius from one circle and a radius from the other with no gap and no overlap.
A diameter is the whole way straight across a circle through its center, and the center splits it into two equal radii that add back to the whole.
Since the circles are all the same size, every radius is the same length, so 'radius plus radius' for the side and 'radius plus radius' for the diameter are the very same total, and two lengths equal to the same total are equal to each other.
The triangle is equilateral, so all three sides are equal diameters. The perimeter is three diameters.
Perimeter adds up the three equal sides, each one diameter long.
3.MD.D.8Identify SubproblemsThree diameters make 36 inches, so divide by 3: 36 ÷ 3 = 12 in.
Sharing 36 equally among 3 equal sides is division within 100.
3.OA.C.7Identify SubproblemsEach side between two touching circles is one diameter -- divide the perimeter by 3 with Grade 3 division!