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Circle · overlapping quadrants in a square
P6 · Circles · PSLE Maths
Question
Ali designed a pattern on a square card ABCD of side 14 cm. He drew two quadrants inside the square, with centres A and C respectively. Both quadrants pass through B and D. The overlapping region of the two quadrants is shaded. Taking π = 3.14, find the area and perimeter of the shaded region.
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Hint
Think first: the two quadrants together cover the whole square. How many times is the shaded part counted?
Show the answer and step-by-step explanation
Answer: Area of shaded region 111.72 cm²; Perimeter of shaded region 43.96 cm
- Read the diagram — The square has side 14 cm. A quadrant is drawn from centre A and another from centre C. Their overlap is the shaded region.
- One quadrant — The radius of each quadrant is the side of the square: 14 cm. Find the area of one quadrant first.
- Add two quadrants — The area of the two quadrants is 2 × 153.86. Notice that the shaded overlap is counted twice.
- Subtract the square — The two quadrants together cover the whole square. So: area of two quadrants = area of square + shaded area.
- Count the arcs — The shaded perimeter has no straight sides. It is made up of 2 quadrant arcs. Each arc is one quarter of a circumference.
- Final answer — The area of the shaded region is 111.72 cm², and its perimeter is 43.96 cm. 📌 Key idea: for overlapping figures, add the larger parts first, then subtract what should be counted only once. For perimeter, count only the outer boundary.
Method: Add the areas of the two quadrants. If their union is exactly the square, then area of two quadrants = area of square + area of overlap. For the perimeter, count only the arcs on the boundary of the shaded region.
Practice: 3 similar questions
- In a square of side 10 cm, two quadrants are drawn with centres at a pair of opposite vertices. Both quadrants pass through the other two vertices. The overlapping part is shaded. Taking π = 3.14, find the area and perimeter of the shaded region.
Answer
57 cm², 31.4 cm — Area of one quadrant = 1/4 × 3.14 × 10 × 10 = 78.5 cm². Area of two quadrants = 2 × 78.5 = 157 cm². Area of the square = 10 × 10 = 100 cm², so shaded area = 157 − 100 = 57 cm². Perimeter of the shaded region = 2 quadrant arcs = 10π = 31.4 cm.
- Siti stitched a design on a square cloth of side 20 cm. She drew two quadrants with centres at two opposite corners. Both quadrants pass through the other two corners. The overlapping part is the design. Taking π = 3.14, find the area and perimeter of the design.
Answer
228 cm², 62.8 cm — Area of one quadrant = 1/4 × 3.14 × 20 × 20 = 314 cm². Area of two quadrants = 2 × 314 = 628 cm². Area of the square = 20 × 20 = 400 cm². Area of the design = 628 − 400 = 228 cm². Perimeter of the design = 2 quadrant arcs = 20π = 62.8 cm.
- Raju has a square sticker of side 28 cm. At each of the four corners, he draws a quadrant of radius 14 cm, with arcs joining the midpoints of adjacent sides. The middle part not covered by the four quadrants is shaded. Taking π = 3.14, find the area and perimeter of the shaded region.
Answer
168.56 cm², 87.92 cm — Area of the square = 28 × 28 = 784 cm². The four quadrants make one circle of radius 14 cm, with area = 3.14 × 14 × 14 = 615.44 cm². Shaded area = 784 − 615.44 = 168.56 cm². The shaded perimeter is made up of 4 quadrant arcs, which is one full circumference: 2 × 3.14 × 14 = 87.92 cm.
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