| a) A = π(a/2)² - 1/2 π(a/2)² + 1/2 (a² - π(a/2)²) = 1/2 a² |
| u = 2 π (a/2) = a π |
| |
| b) A = 1/4 πa² - (a² - 1/4 πa²) = 1/2 πa² - a² = a²(π/2 -1) |
| u = 1/2 * 2 πa = a π |
| |
| c) A = π(a/2)² + 1/4 πa² - 1/2 a² = 1/2 πa² - 1/2 a² = a²/2 (π - 1) |
| u = 2 π(a/2) + 1/4 * 2 πa + 2a + √(2a²) = 3/2 πa + 2a + a√2 |
| u = a(3/2 π + 2 + √2) |
| |
| d) A = 1/2 πa² - (1/4 π 2a² - 1/2 * 2a * a) = a² |
| u = πa + 1/4 * 2 π √(2a²) = πa + a/2 π √2 = a π(1 + (√2)/2) |
| |
| e) A = 1/4 πa² - 3/4 π(a/2)² + ((a/2)² - 1/4 π(a/2)²) |
| A = a²/4 π - 3a²/16 π + a²/4 - a²/16 π = a²/4 |
| u = 1/4 * 2 πa + 3/4 * 2 π a/2 + 2(a/2) = a/2 π + 3a/4 π + a |
| u = a(5/4 π + 1) |
| |
| f) Kleiner Radius: a/2 |
| Großer Radius: r = √(a² - (a/2)²) (Pythagoras) |
| r = a/2 √3 |
| Winkel im gleichseitigen Dreieck: 60° |
| 60°/360° = 1/6 |
| A = 1/6 * 2 π(a/2 √3)² - 1/6 * 2 π(a/2)² |
| A = a²/4 π - a²/12 π = a²/6 π |
| u = 1/6 * 2 π (a/2 √3) + 1/6 * 2 π a/2 + 2(a/2 √3 - a/2) |
| u = a/6 π √3 + a/6 π + a √3 - a = a(π/6 √3 + π/6 + √3 - 1) |
| |
| g) Radius: r = 1/2 √(2a²) (Pythagoras) |
| r = a/2 √2 |
| A = 8(1/4 π (a/2 √2)² - 1/2 a * a/2) = 8(a²/8 π - a²/4) |
| A = a² π - 2a² = a²(π -2) |
| u = 8(1/4 * 2 π(a/2 √2)) = 2a π √2 = 2 π √2 a |
| |
| h) Höhe im Dreieck: h = √(a² - (a/2)²) = a/2 √3 |
| A = 1/2 (a/2 √3) a - 3(1/6 π(a/2)²) = a²/4 √3 - a²/8 π |
| A = a²(1/4 √3 - 1/8 π) |
| u = 3(1/6 * 2 π a/2) = π/2 a |
| |
| i) A = πa² - π(a/2)² + 1/6 π(a/2)² = πa² - πa²/4 + πa²/24 = 19/24 πa² |
| u = 2 πa + 5/6 * 2 πa/2 + 2 a/2 = a(2 π + 5/6 π + 1) = a(17/6 π + 1) |
| |
| k) A = 1/2 π(5/2 a)² - 1/2 π(3/2 a)² + 1/2 (5a + 3a) * 5/2 a |
| A = 25a²/8 π - 9a²/8 π + 10a² = 2a² π + 10a² = a²(2 π + 10) |
| u = 1/2 * 2 π(5/2 a) + 1/2 * 2 π(3/2 a) + 3a + 3a + 2(√(a² + (5/2 a)²)) |
| u = 4a π + 6a + 2(√(29/4 a²)) = a(4 π + 6 + √(29)) |
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