கணிதம் உங்களுக்காக - 13 - கான்ஸெப்ட்கள்

 1. Volume of a regular dodecahedron: V = (15 + 7√5) / 4 a³ (where a is the side length)

2. Surface area of a regular dodecahedron: SA = 3√25 + 10√5 a² (where a is the side length)

3. Volume of a regular icosahedron: V = 5(3 + √5) / 12 a³ (where a is the side length)

4. Surface area of a regular icosahedron: SA = 5√3 a² (where a is the side length)

5. Distance from a point to a line in 3D: d = |Ax₁ + By₁ + Cz₁ + D| / √(A² + B² + C²)

6. Distance between two points in 3D: d = √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²)

7. Midpoint in 3D: M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2, (z₁ + z₂) / 2)

8. Standard form of a parabola (vertical): y - k = a(x - h)² (where (h, k) is the vertex)

9. Standard form of a parabola (horizontal): x - h = a(y - k)² (where (h, k) is the vertex)

10. Standard form of an ellipse (centered at (h, k)): (x - h)² / a² + (y - k)² / b² = 1

11. Standard form of a hyperbola (horizontal): (x - h)² / a² - (y - k)² / b² = 1

12. Standard form of a hyperbola (vertical): (y - k)² / a² - (x - h)² / b² = 1

13. Dot product: A · B = A₁B₁ + A₂B₂ + A₃B₃

14. Vector cross product: A × B = (A₂B₃ - A₃B₂)i + (A₃B₁ - A₁B₃)j + (A₁B₂ - A₂B₁)k

15. Magnitude of a vector in 3D: |A| = √(A₁² + A₂² + A₃²)

16. Scalar projection of vector A onto vector B: proj_B(A) = (A · B) / |B|

17. Vector projection of vector A onto vector B: Proj_B(A) = ((A · B) / |B|²) B

18. Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)

19. Law of Cosines: c² = a² + b² - 2ab cos(C)

20. Law of Tangents: (a - b) / (a + b) = tan[(A - B)/2] / tan[(A + B)/2]

21. Area of a triangle using trigonometry: A = ½ab sin(C)

22. Polar to Cartesian coordinates: x = r cos(θ), y = r sin(θ)

23. Cartesian to Polar coordinates: r = √(x² + y²), θ = tan⁻¹(y / x)

24. 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)

25. Segment lengths in a circle: If two chords AB and CD intersect at point P inside the circle, then PA × PB = PC × PD.

26. Length of a tangent segment: If a tangent from an external point P touches the circle at T, then PT² = PA × PB (where A and B are points of intersection of a secant through P)

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

28. Area of a sector: A = ½r²θ (where r is the radius and θ is the central angle in radians)

29. Segment area: A = ½r²(θ - sinθ)

30. Length of a median of a triangle: m = ½ √(2b² + 2c² - a²) (where a, b, and c are the lengths of the sides of the triangle, and m is the median to side a)

31. Length of an altitude in a triangle: h = 2A / a (where A is the area of the triangle and a is the base)

32. Length of an angle bisector in a triangle: l = √(bc[1 - (a² / (b + c)²)]) (where a, b, and c are the lengths of the sides of the triangle, and l is the angle bisector)

33. Area of a cyclic quadrilateral: A = √((s - a)(s - b)(s - c)(s - d)) (where a, b, c, and d are the lengths of the sides, and s is the semi-perimeter)

34. 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)

35. Area of an inscribed circle in a triangle: A = r × s (where r is the radius of the inscribed circle and s is the semi-perimeter)