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Geometry

Study it inGeometry Level 1: Shapes and Space

Map

Arrows point from what you learn first to what it opens. Point at a concept to light up its chain and where it comes back deeper. Right-click a concept (or press and hold it) for its lessons and stories.

  • Proven to depth123
  • Open
  • Locked
  • Comes back deeper
Depth 3Depth 2Depth 2Depth 2Depth 2AnglesAreaCirclesCoordinatesParallel and perpendicular linesPerimeterPi (π)Points and linesPolygonsPythagoras' theoremQuadrilateralsSimilar shapesSlides, turns and flipsSolid shapesSymmetryTrianglesLengthIn MeasurementThe number lineIn Number SystemsAdditionIn ArithmeticDecimalsIn FractionsSquare numbersIn Number TheorySquare rootsIn ArithmeticMultiplicationIn Arithmetic

Learning path

In the order the concepts build on each other.

  1. Points and linesLockedA point is a place, with no size. A line is straight and goes on for ever in both directions, a line segment is a piece of a line with two ends, and a ray starts at a point and goes on for ever in one direction. The straight path between two points is the shortest, and only one straight line passes through both.Opens after: Length
  2. AnglesLockedAn angle is the amount of turn between two lines that start at the same point. A quarter turn is a right angle, like the corner of a page. Angles are measured in degrees: 360° make a full turn, a right angle is 90° and a straight angle 180°. Angles smaller than 90° are acute, and those between 90° and 180° are obtuse.Opens after: Points and lines
  3. CirclesLockedA circle is every point at the same distance from one point, its centre. That distance is the radius; the diameter goes across through the centre and is two radii; the circumference is the way round. A circle's area is π × radius × radius.Opens after: Points and lines, Length
  4. Parallel and perpendicular linesLockedParallel lines stay the same distance apart and never meet, like the two rails of a railway. Perpendicular lines meet or cross at a right angle, like the edges at the corner of a page. Two lines that are both perpendicular to a third line are parallel to each other.Opens after: Angles
  5. PolygonsLockedA polygon is a flat shape closed by straight sides that meet only at their ends, like a triangle or a stop sign. It is named by how many sides it has (a pentagon has 5, a hexagon 6, an octagon 8), and it has as many vertices (corners) as sides. A regular polygon has all its sides equal and all its angles equal.Opens after: Points and lines, Angles
  6. CoordinatesLockedTwo numbers name a point on a grid: how far across from the origin, then how far up, written (3, 2). The two number lines are the axes and they meet at the origin, (0, 0). The order matters: (3, 2) and (2, 3) are different points. With coordinates a shape becomes a list of numbers.Opens after: The number line, Parallel and perpendicular lines
  7. PerimeterLockedThe perimeter of a shape is the whole distance round its edge: add up the lengths of all its sides. For a rectangle, add the length and the width and double it. It is a length, so it is measured in cm or m.Opens after: Polygons, Length, Addition
  8. QuadrilateralsLockedA quadrilateral is a polygon with four sides. Its family includes the rectangle (four right angles), the square (four right angles and four equal sides), the parallelogram (two pairs of parallel sides), the rhombus (four equal sides), the trapezium (one pair of parallel sides) and the kite (two pairs of equal sides next to each other). A square is a rectangle and a rhombus at once. A diagonal joins two opposite vertices and cuts a quadrilateral into two triangles, so its angles add up to 360°.Opens after: Polygons, Parallel and perpendicular lines
  9. Similar shapesLockedTwo shapes are similar when they have the same shape but maybe different sizes: every length of one is the same number of times the matching length of the other, and every angle is the same. Maps, plans and models are drawn to a scale, like 1 to 100. Doubling every length makes the area four times as big.Opens after: Polygons, Angles, Multiplication
  10. Solid shapesLockedA solid shape takes up space: it has a length, a width and a height. Most solids are flat faces that meet at edges and vertices - a box has 6 faces, 12 edges and 8 vertices - and the main families are the cube, cuboid, prism, pyramid, cylinder, cone and sphere. Opened out flat, a solid becomes its net.Opens after: Polygons, Circles
  11. SymmetryLockedA shape has symmetry when a line folds it into two halves that land exactly on each other, each a mirror image of the other. That line is a line of symmetry, or mirror line. A rectangle has 2, a square 4, a regular polygon as many as it has sides, a circle endlessly many, and most parallelograms none.Opens after: Polygons
  12. TrianglesLockedA triangle is a polygon with three sides. By its sides it is equilateral (all three equal), isosceles (two equal) or scalene (none equal); by its angles it can be right-angled, have one obtuse angle, or have all acute angles. Three sides fit together in only one way, so a triangle can't wobble - that is why frames and bridges are built from triangles. Its three angles always add up to 180°.Opens after: Polygons
  13. AreaLockedArea is how much flat surface a shape covers, measured by counting equal squares that cover it with no gaps or overlaps: square centimetres (cm²) or square metres (m²). A rectangle's area is length × width, a parallelogram's is base × height, and a triangle's is half of base × height.Opens after: Quadrilaterals, Length
  14. Pi (π)LockedPi, written π, is the number you get by dividing any circle's circumference by its diameter: about 3.14, the same for every circle. So circumference = π × diameter. Its digits never end and never repeat: π is an irrational number.Opens after: Circles, Perimeter, Decimals
  15. Slides, turns and flipsLockedA shape can be moved without changing it in three ways: a slide (translation) moves every point the same distance in the same direction, a turn (rotation) spins it about a fixed point by an angle, and a flip (reflection) reflects it in a mirror line. The copy is congruent to the original: the same shape and size. Only a flip changes which way round it is.Opens after: Angles, Symmetry
  16. Pythagoras' theoremLockedIn a right-angled triangle, the square on the longest side (the hypotenuse) has the same area as the squares on the two shorter sides together: for sides of 3, 4 and 5, 9 + 16 = 25. So two sides give you the third, and a triangle whose sides fit the rule always has a right angle.Opens after: Triangles, Area, Square numbers, Square roots

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