கணிதம் உங்களுக்காக - 14 - கான்ஸெப்ட்கள்

 1. Volume of a frustum of a 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)

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

3. Volume of a triangular prism: V = 1/2bhL (where b is the base length, h is the height, and L is the length of the prism)

4. Surface area of a triangular prism: SA = bh + 2ls + lb (where b is the base length, h is the height, l is the length, and s is the slant height)

5. Volume of a parallelepiped: V = |a · (b × c)| (where a, b, and c are vectors representing the edges)

6. Equation of a plane: Ax + By + Cz + D = 0

7. Distance from a point to a plane: d = |Ax₁ + By₁ + Cz₁ + D| / √(A² + B² + C²)

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

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

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

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

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

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

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

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

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

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

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

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

20. Area of a triangle using trigonometry: A = 1/2ab sin(C)

21. 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)

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

23. 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)

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

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

26. Segment area: A = 1/2r²(θ - sinθ)

27. Length of a median of a triangle: m = 1/2 √(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)

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

29. 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)

30. 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)

31. Brahmagupta's formula (area of a cyclic quadrilateral): A = √((s - a)(s - b)(s - c)(s - d) - abcd cos²(1/2θ)) (where θ is the sum of the opposite angles)

32. 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)