Russia 11 класс Geometry (advanced)
Chapters: 3
1. Solids of revolution
Cylinder, cone, sphere · Volumes · Combinations of solids · Surface area and similarity · Sections
- Solids of Revolution – A solid of revolution is the 3D shape you get when a flat shape turns a full 360° around a straight line (the axis). A rectangle makes a cylinder, a right triangle makes a cone, a half-circle makes a sphere and a trapezium makes a frustum. Their volumes are V = πr²h, V = ⅓πr²h, V = ⁴⁄₃πr³ and V = ⅓πh(R² + Rr + r²). In calculus, any curve y = f(x) turned about the x-axis gives V = π∫y² dx.
2. Vectors and coordinates
Vector product · Coordinate-vector method
- Vector Algebra – A vector has a size (magnitude) and a direction. In 3D we write it as a = xî + yĵ + zk̂. Its length is |a| = √(x² + y² + z²). Its direction cosines are l = x/|a|, m = y/|a|, n = z/|a|, and l² + m² + n² = 1. Vectors are added head-to-tail (triangle law) or component by component. ka stretches a by k and flips it if k is negative. The point dividing AB in m : n has position vector (mb + na)/(m + n) inside and (mb − na)/(m − n) outside. Dot product a·b = |a||b|cosθ gives a number and tells the angle and the projection. Cross product a×b = |a||b|sinθ n̂ gives a vector at right angles to both; its length is the area of the parallelogram on a and b.
- Three Dimensional Geometry – A line in space is fixed by one point on it and its direction. Its direction cosines l, m, n satisfy l² + m² + n² = 1; any numbers in the same ratio are direction ratios, and through two points they are (x₂ − x₁, y₂ − y₁, z₂ − z₁). Vector equation: r = a + λb. Cartesian equation: (x − x₁)/a = (y − y₁)/b = (z − z₁)/c. The angle between two lines is the angle between their directions: cosθ = |b₁·b₂|/(|b₁||b₂|). Two lines in space are parallel, intersecting or skew. The shortest distance between skew lines is d = |(a₂ − a₁)·(b₁ × b₂)|/|b₁ × b₂|; for parallel lines d = |b × (a₂ − a₁)|/|b|; d = 0 means they meet.
3. Motions in space
Motions
- Transformations: Slide, Flip, Turn and Resize a Shape – A transformation moves a shape to a new place. Translation slides it, reflection flips it in a mirror line, rotation turns it about a centre, enlargement changes its size from a centre. The first three keep size and shape (congruent image). Enlargement keeps shape but changes size (similar image). Each has a simple coordinate rule.