பென் டென் - தொடரின் கடந்த கால ஏலியன்கள்

1. Heatblast - Pyronite: Can generate and control fire, absorb heat, and fly using fire propulsion.

2. Wildmutt - Vulpimancer: Enhanced senses, agility, and strength; can track scents and see in the dark.

3. Diamondhead - Petrosapien: Can generate and manipulate crystal, create weapons and shields, and regenerate damaged crystal.

4. XLR8 - Kineceleran: Super speed, agility, and reflexes; can create tornadoes by running in circles.

5. Grey Matter - Galvan: Super intelligence, can fit into small spaces, and has enhanced agility.

6. Four Arms - Tetramand: Super strength, durability, and can create shockwaves by clapping hands.

7. Stinkfly - Lepidopterran: Can fly, shoot slime, and has enhanced agility and strength.

8. Ripjaws - Piscciss Volann: Can breathe underwater, swim at high speeds, and has powerful jaws and claws.

9. Upgrade - Galvanic Mechamorph: Can merge with and upgrade technology, shoot energy beams, and shape-shift.

10. Ghostfreak - Ectonurite: Can become intangible, invisible, and possess other beings.

11. Cannonbolt - Arburian Pelarota: Can roll into a ball, has super strength and durability.

12. Wildvine - Florauna: Can stretch limbs, grow vines, and regenerate.

13. Spitter - Spheroid: Can spit slime that can trap enemies and create slippery surfaces.

14. Buzzshock - Nosedeenian: Can generate and control electricity, and travel through electrical currents.

15. Arctiguana - Polar Manzardill: Can generate and control ice, and has enhanced strength and durability.

16. Upchuck - Gourmand: Can eat and digest any material, and shoot explosive projectiles.

17. Way Big - To'kustar: Can grow to enormous size, has super strength, and can shoot cosmic rays.

18. Big Chill - Necrofriggian: Can become intangible, fly, and generate extreme cold.

19. Echo Echo - Sonorosian: Can create sonic blasts, duplicate himself, and merge duplicates.

20. Humungousaur - Vaxasaurian: Can grow in size, has super strength, and durability.

21. Jetray - Aerophibian: Can fly at high speeds, shoot neuroshock blasts, and breathe underwater.

22. Chromastone - Crystalsapien: Can absorb and redirect energy, and has enhanced strength and durability.

23. Brainstorm - Cerebrocrustacean: Super intelligence, can generate and control electricity, and has telekinesis.

24. Alien X - Celestialsapien: Can alter reality, has invulnerability, and can regenerate.

25. Clockwork - Chronosapien: Can control time, travel through time, and alter timelines.

26. Atomix - Unknown species: Can generate atomic energy, create nuclear blasts, and fly.

Additional Aliens

1. Swampfire - Methanosian: Can generate and control fire, regenerate, and has super strength.

2. Ultimate Humungousaur - Enhanced version of Humungousaur: Can grow bone-like armor and shoot missiles.

3. Shocksquatch - Gimlinopithecus: Can generate and control electricity.

4. Overflow - Cascan: Can shoot water jets and create water constructs.

5. Slapback - Ekoplektoid: Can duplicate himself and each duplicate becomes smaller and stronger.

கணிதம் உங்களுக்காக - 37 - கான்ஸெப்ட்கள்

 

1. General form of the equation of a circle: x² + y² + Dx + Ey + F = 0 (where D, E, and F are constants)

2. Distance between two points in a plane: d = √((x₂ - x₁)² + (y₂ - y₁)²)

3. Midpoint formula: M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)

4. Slope of a line passing through points (x₁, y₁) and (x₂, y₂): m = (y₂ - y₁) / (x₂ - x₁)

5. Equation of a line in slope-intercept form: y = mx + c (where m is the slope and c is the y-intercept)

6. Equation of a line in point-slope form: y - y₁ = m(x - x₁)

7. General form of a line equation: Ax + By + C = 0

8. Distance from a point to a line: d = |Ax₁ + By₁ + C| / √(A² + B²) (where (x₁, y₁) is the point and Ax + By + C = 0 is the line)

9. Volume of a regular tetrahedron: V = (√2 / 12) a³ (where a is the side length)

10. Surface area of a regular tetrahedron: SA = √3 a² (where a is the side length)

11. Volume of an octahedron: V = (√2 / 3) a³ (where a is the side length)

12. Surface area of an octahedron: SA = 2√3 a² (where a is the side length)

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

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

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

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

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

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

19. Polar form of the equation of a line: r = l / cos(θ - φ) (where l is the perpendicular distance from the origin and φ is the angle with the positive x-axis)

20. Polar form of the equation of a circle: r(θ) = R (where R is the radius)

21. Parametric equations for a circle in polar coordinates: x = R cos(θ), y = R sin(θ) (where R is the radius and θ is the parameter)

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

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

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

25. Angle between two vectors: cos(θ) = (A · B) / (|A||B|)

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

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

28. Product-to-Sum Formulas:

    - sin(A) sin(B) = ½ [cos(A - B) - cos(A + B)]

    - cos(A) cos(B) = ½ [cos(A + B) + cos(A - B)]

    - sin(A) cos(B) = ½ [sin(A + B) + sin(A - B)]

29. Sum-to-Product Formulas:

    - sin(A) ± sin(B) = 2 sin((A ± B) / 2) cos((A ∓ B) / 2)

    - cos(A) + cos(B) = 2 cos((A + B) / 2) cos((A - B) / 2)

    - cos(A) - cos(B) = -2 sin((A + B) / 2) sin((A - B) / 2)

30. Half-Angle Formulas:

    - sin²(A/2) = (1 - cosA) / 2

    - cos²(A/2) = (1 + cosA) / 2

    - tan²(A/2) = (1 - cosA) / (1 + cosA)

31. Double Angle Formulas:

    - sin(2A) = 2 sinA cosA

    - cos(2A) = cos²A - sin²A = 2 cos²A - 1 = 1 - 2 sin²A

    - tan(2A) = 2 tanA / (1 - tan²A)

32. Triple Angle Formulas:

    - sin(3A) = 3 sinA - 4 sin³A

    - cos(3A) = 4 cos³A - 3 cosA

    - tan(3A) = (3 tanA - tan³A) / (1 - 3 tan²A)

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)

கணிதம் உங்களுக்காக - 35 - கான்ஸெப்ட்கள்

 

1. Equation of an ellipse in standard form (centered at the origin): (x² / a²) + (y² / b²) = 1 (where a is the semi-major axis and b is the semi-minor axis)

2. Equation of a hyperbola in standard form (centered at the origin): (x² / a²) - (y² / b²) = 1 (horizontal) or (y² / a²) - (x² / b²) = 1 (vertical)

3. Asymptotes of a hyperbola: y = ±(b / a)x (horizontal) or y = ±(a / b)x (vertical)

4. Parametric equations of a circle: x = h + r cos(t), y = k + r sin(t) (where (h, k) is the center, r is the radius, and t is the parameter)

5. Parametric equations of an ellipse: x = h + a cos(t), y = k + b sin(t) (where (h, k) is the center, a is the semi-major axis, b is the semi-minor axis, and t is the parameter)

6. Polar form of the equation of a line: r = (l) / (cos(θ - φ)) (where l is the perpendicular distance from the origin and φ is the angle with the positive x-axis)

7. Polar form of the equation of a circle: r(θ) = R (where R is the radius)

8. Volume of a spherical cap: V = (1/6)πh(3a² + 3b² + h²) (where h is the height of the cap and a and b are the radii of the cap's bases)

9. Surface area of a spherical cap: SA = 2πRh (where h is the height of the cap and R is the radius of the sphere)

10. Volume of a spherical sector: V = (2/3)πR²h (where R is the radius of the sphere and h is the height of the spherical sector)

11. Volume of a spherical zone: V = (2/3)πR²(h₂ - h₁) (where R is the radius of the sphere, h₁ is the height of the lower zone, and h₂ is the height of the upper zone)

12. Surface area of a spherical zone: SA = 2πRh (where R is the radius of the sphere and h is the height of the zone)

13. Volume of a spherical wedge: V = (2/3)πR²θ (where R is the radius of the sphere and θ is the dihedral angle)

14. Surface area of a spherical wedge: SA = 2πRh (where R is the radius of the sphere and h is the height of the spherical wedge)

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

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

17. 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)

18. Equation of a line in vector form: r = r₀ + t(v) (where r is the position vector, r₀ is a point on the line, v is the direction vector, and t is a scalar)

19. Equation of a plane in vector form: r · n = d (where r is the position vector, n is the normal vector, and d is the distance from the origin)

20. Equation of a sphere in vector form: |r - r₀| = R (where r is the position vector, r₀ is the center, and R is the radius)

21. 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)

22. 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)

23. 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)

24. Length of the altitude of a right triangle: h = (a * b) / c (where a and b are the legs and c is the hypotenuse)

25. Length of the altitude of an equilateral triangle: h = (√3 / 2) * a (where a is the side length)

26. Length of the median of a trapezoid: m = (a + b) / 2 (where a and b are the lengths of the parallel sides)

27. Length of the diagonal of a rectangle: d = √(a² + b²) (where a and b are the side lengths)

28. Length of the diagonal of a parallelogram: d = √(a² + b² + 2ab cos(θ)) (where a and b are the side lengths and θ is the angle between the sides)

29. Length of the side of a regular polygon: s = 2R sin(π / n) (where R is the radius of the circumscribed circle and n is the number of sides)

30. Radius of the circumscribed circle of a regular polygon: R = (s / 2) * csc(π / n) (where s is the side length and n is the number of sides)

31. Radius of the inscribed circle of a regular polygon: r = (s / 2) * cot(π / n) (where s is the side length and n is the number of sides)

32. Product-to-Sum Formulas:

    - sin(A) sin(B) = ½ [cos(A - B) - cos(A + B)]

    - cos(A) cos(B) = ½ [cos(A + B) + cos(A - B)]

    - sin(A) cos(B) = ½ [sin(A + B) + sin(A - B)]

33. Sum-to-Product Formulas:

    - sin(A) ± sin(B) = 2 sin((A ± B) / 2) cos((A ∓ B) / 2)

    - cos(A) + cos(B) = 2 cos((A + B) / 2) cos((A - B) / 2)

    - cos(A) - cos(B) = -2 sin((A + B) / 2) sin((A - B) / 2)

34. Half-Angle Formulas:

    - sin²(A/2) = (1 - cosA) / 2

    - cos²(A/2) = (1 + cosA) / 2

    - tan²(A/2) = (1 - cosA) / (1 + cosA)

35. Double Angle Formulas:

    - sin(2A) = 2 sinA cosA

    - cos(2A) = cos²A - sin²A = 2 cos²A - 1 = 1 - 2 sin²A

    - tan(2A) = 2 tanA / (1 - tan²A)

36. Triple Angle Formulas:

    - sin(3A) = 3 sinA - 4 sin³A

    - cos(3A) = 4 cos³A - 3 cosA

    - tan(3A) = (3 tanA - tan³A) / (1 - 3 tan²A)

கணிதம் உங்களுக்காக - 34 - கான்ஸெப்ட்கள்

1. Volume of a right circular cone: V = (1/3)πr²h (where r is the radius of the base and h is the height)

2. Surface area of a right circular cone: SA = πr(r + √(r² + h²))

3. Volume of a right circular cylinder: V = πr²h (where r is the radius of the base and h is the height)

4. Surface area of a right circular cylinder: SA = 2πrh + 2πr²

5. Volume of a spherical segment: V = (1/6)πh(3a² + 3b² + h²) (where h is the height of the segment, a and b are the radii of the segment's bases)

6. Surface area of a spherical segment: SA = 2πRh (where R is the radius of the sphere and h is the height of the segment)

7. Length of the altitude of a right triangle: h = (a * b) / c (where a and b are the legs and c is the hypotenuse)

8. Length of the altitude of an equilateral triangle: h = (√3 / 2) * a (where a is the side length)

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

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

11. Length of the median of a trapezoid: m = (a + b) / 2 (where a and b are the lengths of the parallel sides)

12. Length of the diagonal of a rectangle: d = √(a² + b²) (where a and b are the side lengths)

13. Length of the diagonal of a parallelogram: d = √(a² + b² + 2ab cos(θ)) (where a and b are the side lengths and θ is the angle between the sides)

14. Length of the side of a regular polygon: s = 2R sin(π / n) (where R is the radius of the circumscribed circle and n is the number of sides)

15. Radius of the circumscribed circle of a regular polygon: R = (s / 2) * csc(π / n) (where s is the side length and n is the number of sides)

16. Radius of the inscribed circle of a regular polygon: r = (s / 2) * cot(π / n) (where s is the side length and n is the number of sides)

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

18. Equation of the ellipse (standard form): (x - h)² / a² + (y - k)² / b² = 1 (where (h, k) is the center, a is the semi-major axis, and b is the semi-minor axis)

19. Equation of the hyperbola (standard form): (x - h)² / a² - (y - k)² / b² = 1 (horizontal) or (y - k)² / a² - (x - h)² / b² = 1 (vertical)

20. Slopes of the asymptotes of a hyperbola: ±(b / a) (horizontal) or ±(a / b) (vertical)

21. Distance between two parallel lines: d = |c₂ - c₁| / √(a² + b²) (where the equations of the lines are Ax + By + C₁ = 0 and Ax + By + C₂ = 0)

22. Distance between a point and a plane: d = |Ax₁ + By₁ + Cz₁ + D| / √(A² + B² + C²)

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

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

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

26. 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)

27. Product-to-Sum Formulas:

    - sin(A) sin(B) = ½ [cos(A - B) - cos(A + B)]

    - cos(A) cos(B) = ½ [cos(A + B) + cos(A - B)]

    - sin(A) cos(B) = ½ [sin(A + B) + sin(A - B)]

28. Sum-to-Product Formulas:

    - sin(A) ± sin(B) = 2 sin((A ± B) / 2) cos((A ∓ B) / 2)

    - cos(A) + cos(B) = 2 cos((A + B) / 2) cos((A - B) / 2)

    - cos(A) - cos(B) = -2 sin((A + B) / 2) sin((A - B) / 2)

29. Half-Angle Formulas:

    - sin²(A/2) = (1 - cosA) / 2

    - cos²(A/2) = (1 + cosA) / 2

    - tan²(A/2) = (1 - cosA) / (1 + cosA)

30. Double Angle Formulas:

    - sin(2A) = 2 sinA cosA

    - cos(2A) = cos²A - sin²A = 2 cos²A - 1 = 1 - 2 sin²A

    - tan(2A) = 2 tanA / (1 - tan²A)

31. Triple Angle Formulas:

    - sin(3A) = 3 sinA - 4 sin³A

    - cos(3A) = 4 cos³A - 3 cosA

    - tan(3A) = (3 tanA - tan³A) / (1 - 3 tan²A)

கணிதம் உங்களுக்காக - 33 - கான்ஸெப்ட்கள்

 1. Equation of a straight line in general form: Ax + By + C = 0

2. Equation of a line in slope-intercept form: y = mx + c (where m is the slope and c is the y-intercept)

3. Distance between two points (x₁, y₁) and (x₂, y₂): d = √((x₂ - x₁)² + (y₂ - y₁)²)

4. Midpoint formula: M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)

5. Slope of a line through points (x₁, y₁) and (x₂, y₂): m = (y₂ - y₁) / (x₂ - x₁)

6. Distance from a point to a line Ax + By + C = 0: d = |Ax₁ + By₁ + C| / √(A² + B²)

7. Equation of a circle in standard form: (x - h)² + (y - k)² = r² (where (h, k) is the center and r is the radius)

8. Equation of a parabola in standard form: y = ax² + bx + c or (y - k) = a(x - h)² (where (h, k) is the vertex)

9. Volume of a frustum of a cone: V = (1/3)πh(R₁² + R₁R₂ + R₂²) (where h is the height, R₁ is the radius of the top base, and R₂ is the radius of the bottom base)

10. Surface area of a frustum of a cone: SA = π(R₁ + R₂)√((R₁ - R₂)² + h²) + πR₁² + πR₂²

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

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

13. Volume of a paraboloid: V = (1/2)πr²h

14. 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)

15. 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)

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

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

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

19. Half-Angle Formulas:

    - sin²(A/2) = (1 - cosA) / 2

    - cos²(A/2) = (1 + cosA) / 2

    - tan²(A/2) = (1 - cosA) / (1 + cosA)

20. Double Angle Formulas:

    - sin(2A) = 2 sinA cosA

    - cos(2A) = cos²A - sin²A = 2 cos²A - 1 = 1 - 2 sin²A

    - tan(2A) = 2 tanA / (1 - tan²A)

21. Triple Angle Formulas:

    - sin(3A) = 3 sinA - 4 sin³A

    - cos(3A) = 4 cos³A - 3 cosA

    - tan(3A) = (3 tanA - tan³A) / (1 - 3 tan²A)

22. Product-to-Sum Formulas:

    - sin(A) sin(B) = ½ [cos(A - B) - cos(A + B)]

    - cos(A) cos(B) = ½ [cos(A + B) + cos(A - B)]

    - sin(A) cos(B) = ½ [sin(A + B) + sin(A - B)]

23. Sum-to-Product Formulas:

    - sin(A) ± sin(B) = 2 sin((A ± B) / 2) cos((A ∓ B) / 2)

    - cos(A) + cos(B) = 2 cos((A + B) / 2) cos((A - B) / 2)

    - cos(A) - cos(B) = -2 sin((A + B) / 2) sin((A - B) / 2)

கணிதம் உங்களுக்காக - 32 - கான்ஸெப்ட்கள்

1. Length of a chord of a circle: c = 2r sin(θ/2) (where r is the radius and θ is the central angle in radians)

2. Equation of a circle in parametric form: x = h + r cos(t), y = k + r sin(t) (where (h, k) is the center, r is the radius, and t is the parameter)

3. Equation of an ellipse in parametric form: x = h + a cos(t), y = k + b sin(t) (where (h, k) is the center, a is the semi-major axis, b is the semi-minor axis, and t is the parameter)

4. Equation of a hyperbola in parametric form: x = h + a sec(t), y = k + b tan(t) (where (h, k) is the center, a is the semi-major axis, b is the semi-minor axis, and t is the parameter)

5. Locus of a point: The set of all points that satisfy a given condition.

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

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

8. 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)

9. 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)

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

11. Surface area of a parallelepiped: SA = 2(ab + bc + ca) (where a, b, and c are the lengths of the edges)

12. Volume of an oblique cylinder: V = πr²h

13. Surface area of an oblique cylinder: SA = 2πrh + 2πr²

14. Volume of a cone: V = 1/3πr²h

15. Surface area of a cone: SA = πr(r + l) (where l is the slant height)

16. Length of the diagonal of a regular polygon: d = √(2r²(1 - cos(2π/n))) (where r is the radius of the circumscribed circle and n is the number of sides)

17. Radius of the circumscribed circle of a regular polygon: R = (s / 2) * csc(π / n) (where s is the side length and n is the number of sides)

18. Radius of the inscribed circle of a regular polygon: r = (s / 2) * cot(π / n) (where s is the side length and n is the number of sides)

19. Length of a median of a trapezoid: m = (a + b) / 2 (where a and b are the lengths of the parallel sides)

20. Length of the diagonal of a parallelogram: d = √(a² + b² + 2ab cos(θ)) (where a and b are the side lengths and θ is the angle between the sides)

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

22. Law of Cotangents: (a - b) / (a + b) = cot[(A - B)/2] / cot[(A + B)/2]

23. Half-Angle Formulas:

    - sin²(A/2) = (1 - cosA) / 2

    - cos²(A/2) = (1 + cosA) / 2

    - tan²(A/2) = (1 - cosA) / (1 + cosA)

24. Double Angle Formulas:

    - sin(2A) = 2 sinA cosA

    - cos(2A) = cos²A - sin²A = 2 cos²A - 1 = 1 - 2 sin²A

    - tan(2A) = 2 tanA / (1 - tan²A)

25. Triple Angle Formulas:

    - sin(3A) = 3 sinA - 4 sin³A

    - cos(3A) = 4 cos³A - 3 cosA

    - tan(3A) = (3 tanA - tan³A) / (1 - 3 tan²A)

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

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

28. 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)