கணிதம் உங்களுக்காக - 36 - கான்ஸெப்ட்கள்

 

1. Volume of a torus: V = (πr²) × (2πR) = 2π²Rr² (where r is the radius of the tube and R is the distance from the center of the tube to the center of the torus)

2. Surface area of a torus: SA = (2πr) × (2πR) = 4π²Rr

3. Volume of a truncated pyramid: V = (1/3)h(A₁ + A₂ + √(A₁A₂)) (where h is the height, A₁ is the area of the top base, and A₂ is the area of the bottom base)

4. Surface area of a truncated pyramid: SA = A₁ + A₂ + (1/2)Pℓ (where P is the perimeter of the base and ℓ is the slant height)

5. Volume of an ellipsoid: V = (4/3)πabc (where a, b, and c are the semi-axes)

6. Surface area of an ellipsoid (approximate): SA = 4π[(a²b^(p) + a²c^(p) + b²c^(p))/3]^(1/p) (where p ≈ 1.6075)

7. Scalar triple product: V = A · (B × C) (where A, B, and C are vectors)

8. Vector triple product: A × (B × C) = (A · C)B - (A · B)C

9. Volume of a parallelepiped: V = |A · (B × C)| (where A, B, and C are vectors)

10. Distance between two skew lines: d = |(A₁ - A₂) · (B₁ × B₂)| / |B₁ × B₂| (where A₁ and A₂ are points on the lines, and B₁ and B₂ are direction vectors)

11. Heron's formula for the area of a triangle: A = √(s(s - a)(s - b)(s - c)) (where s is the semi-perimeter and a, b, and c are the lengths of the sides)

12. Brahmagupta's formula (area of a cyclic quadrilateral): A = √((s - a)(s - b)(s - c)(s - d) - abcd cos²(½θ)) (where θ is the sum of the opposite angles)

13. Radius of the inscribed circle in a triangle: r = A / s (where A is the area of the triangle and s is the semi-perimeter)

14. Radius of the circumscribed circle in a triangle: R = (abc) / (4A) (where a, b, and c are the side lengths, and A is the area of the triangle)

19. Power of a point theorem: If a point P lies outside a circle and two secant lines PA and PB intersect the circle at A and B, then PA × PB = PC × PD (where C and D are points of intersection)

20. Length of an arc: L = rθ (where r is the radius and θ is the central angle in radians)

21. Area of a sector: A = 1/2r²θ (where r is the radius and θ is the central angle in radians)

22. Segment area: A = 1/2r²(θ - sinθ)

23. Rotation around a point (h, k) by θ degrees: (x', y') = ((x - h) cosθ - (y - k) sinθ + h, (x - h) sinθ + (y - k) cosθ + k)

24. Reflection over a line y = mx + c: (x', y') = ((x(1 - m²) + 2my - 2mc) / (1 + m²), (y(1 - m²) + 2mx + 2c) / (1 + m²))

25. Dilation with respect to a point (h, k): (x', y') = (kx + (1 - k)h, ky + (1 - k)k)

26. Horizontal shear transformation: (x', y') = (x + ky, y)

27. Vertical shear transformation: (x', y') = (x, y + kx)