Area Applications • Topic 2 of 3

Shaded Regions

Shaded-region problems look intimidating because the wanted area is irregular, but almost all of them collapse to one principle: shaded = outer area − inner area. A circle inscribed in a square, a square inside a circle, four corner circles inside a rectangle, a flower-petal pattern — in each case the shaded part is what remains after you subtract one neat region from another. The two recurring CAT setups are worth memorising. For a circle inscribed in a square of side a, the corners left over total a² − π(a/2)² = a²(1 − π/4) ≈ 0.215a². For a square inscribed in a circle of radius r, the square has diagonal 2r so its side is r√2 and area 2r², leaving the four segments. When overlapping circles create lens or petal shapes, build them from sectors minus triangles. The discipline is to name the outer region, name the inner region, and never lose track of which one is shaded.

✅ Solved examples

1. A circle is inscribed in a square of side 14 cm. Find the area outside the circle but inside the square (π ≈ 22/7).
Square = 196. Inscribed circle radius = 7, area = (22/7) × 49 = 154. Shaded corners = 196 − 154 = 42 cm².
2. A square is inscribed in a circle of radius 10 cm. Find the area inside the circle but outside the square. Use π ≈ 3.14.
Circle = π × 100 = 314. Square diagonal = 2r = 20, so side = 20/√2, area = ½ × diagonal² = ½ × 400 = 200. Shaded = 314 − 200 = 114 cm².
3. A rectangle 20 cm by 14 cm has a circle of radius 7 cm cut from its centre. Find the remaining area (π ≈ 22/7).
Rectangle = 280. Circle = (22/7) × 49 = 154. Remaining = 280 − 154 = 126 cm².
4. Four equal circles of radius 7 cm are drawn at the four corners of a square of side 14 cm, each quarter lying inside. Find the area of the square not covered (π ≈ 22/7).
Four quarter-circles = one full circle = (22/7) × 49 = 154. Square = 196. Uncovered = 196 − 154 = 42 cm².

✏️ Practice — try these, take hints as needed

1. A circle of radius 7 cm is inscribed in a square. Find the shaded area between square and circle (π ≈ 22/7).
Square side = 2r = 14.
Square = 196, circle = 154.
Shaded = 196 − 154.
42 cm²
2. A square of side 10 cm has a circle of the largest possible size cut from its centre. Find the remaining area (π ≈ 3.14).
Largest circle radius = 5.
Circle = 3.14 × 25 = 78.5.
100 − 78.5.
21.5 cm²
3. A square is inscribed in a circle of radius 7 cm. Find the area inside the circle but outside the square (π ≈ 22/7).
Square area = ½ × diagonal².
Diagonal = 14, square = 98.
Circle = 154, subtract.
56 cm²
4. A rectangle 16 cm by 9 cm has a circle of radius 4.5 cm cut out. Find the remaining area (π ≈ 22/7).
Rectangle = 144.
Circle = (22/7) × 4.5².
(22/7) × 20.25 ≈ 63.64.
≈ 80.36 cm²
5. Two concentric circles have radii 10 cm and 6 cm. Find the area of the ring between them (π ≈ 3.14).
Ring = π(R² − r²).
R² − r² = 100 − 36 = 64.
3.14 × 64.
200.96 cm²

📝 Topic test — 8 questions

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