For the figures below, assume they are made of semicircles, quarter circles and squares. For each shape, find the area and perimeter. Give your answer as a completely simplified exact value in terms of π (no approximations).
1. Area of the first figure; Area of the quarter circle = 1/4 πr² = 1/4π 12² = 36π cm² Area of the triangle = 1/2×base×height = 1/2 × 12×12 = 72 cm² Thus the area of the region = (36π-72) cm²
2. Perimeter of the first figure The length of the arc AC is 2πr/4 = 2π×12/4 = 6π cm the length of line AC √(12² +12²) = 16.97 Thus the perimeter of the figure = (6π+16.97) cm
3. The are of figure 2 Area of the semi circle πr²/2 = 12²π/2 = 72π cm² The area of the square = 24 × 24 =576 cm² Area of the semi circle = 72 π cm² Thus the area of the part of the square = (576-72π) cm² total area 72π +576-72π = 576 cm²
4. Perimeter of figure 2 The length of arc AD = (2πr)/2 = 2π×12/2 = 12π cm but length of arc AD is equal to that of arc BC Thus arc BC = 12πcm Therefore, total perimeter = 12π +12π +24 +24 = (24π +48) cm
