Pariksha SaathiLesson 6 of 8 · Mathematics, Classes I-V (Paper I)
Geometry and Shapes: Solids, Plane Figures, Symmetry, Tangrams, Perimeter, Area and Nets
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How children come to know shapes
Children handle solids long before they name flat shapes, so NCERT begins with things that roll, slide or stack: balls, boxes, cans and dice (Class 1 Shapes and Space, Class 2 What is Long, What is Round?). Pierre and Dina van Hiele described levels of geometric thinking. At the first level (visualisation) a child recognises a shape by its overall look, so a square tilted onto its corner may be called a diamond; at the next level (analysis) she describes shapes by their properties, such as 4 equal sides and 4 right angles. Sorting, building and drawing move children from the first level to the second.
Solids, faces, edges and vertices
A face is a flat surface, an edge is where two faces meet and a vertex (corner) is where edges meet. A cube and a cuboid each have 6 faces, 12 edges and 8 vertices; a triangular prism has 5 faces, 9 edges and 6 vertices; a square pyramid has 5 faces, 8 edges and 5 vertices. For these flat-faced solids faces + vertices - edges = 2 (Euler's formula), a quick check: 6 + 8 - 12 = 2. A cylinder has 2 flat circular faces, 1 curved surface, 2 curved edges and no vertex; a cone has 1 flat face, 1 curved surface, 1 curved edge and 1 vertex; a sphere has only a curved surface.
Plane shapes and angles
A closed figure made of straight sides is a polygon: a triangle has 3 sides, a quadrilateral 4, a pentagon 5, a hexagon 6 and an octagon 8. A circle has a centre; the radius joins the centre to the circle, and the diameter, which passes through the centre, is twice the radius. Angles are compared with a right angle, the corner of a page (90 degrees): a smaller angle is acute and a larger one, still less than a straight line, is obtuse. A square and a rectangle both have 4 right angles; the square also has 4 equal sides, so every square is a rectangle but not every rectangle is a square.
Symmetry
A figure has line symmetry if it can be folded along a line so that the two halves match exactly; children test it by folding paper and with mirrors. A square has 4 lines of symmetry, a rectangle that is not a square 2, an equilateral triangle 3, an isosceles triangle 1, a scalene triangle none, a regular hexagon 6 and a circle infinitely many. In plain capital letters, A, M, T, U, V, W and Y have a vertical line of symmetry, B, C, D, E and K a horizontal one, and H, I, O and X both. The Class 5 chapter Does it Look the Same? adds turn (rotational) symmetry, in which a shape looks the same after part of a full turn: a square looks the same after every 1/4 turn, a rectangle after a 1/2 turn, and the letters H, N, S, X and Z after a 1/2 turn.
Tangrams and tiling
A tangram is a square cut into 7 pieces (tans): 2 large triangles, 1 medium triangle, 2 small triangles, 1 square and 1 parallelogram. Two small triangles can make the square, the medium triangle or the parallelogram, so these 3 pieces have equal areas, and every picture made from all 7 pieces has the area of the original square: the shape changes, the area does not. Tiling (tessellation) covers a surface without gaps or overlaps; squares, equilateral triangles and regular hexagons tile on their own, regular pentagons and circles do not.
Perimeter and area by counting
Perimeter is the length of the boundary, as of a fence round a field (the Class 4 chapter Fields and Fences); area is the amount of surface, counted in unit squares (the Class 5 chapter Area and its Boundary). On squared paper, count each full square as 1, a part more than half a square as 1 and exactly half a square as 1/2, and ignore parts less than half. The two ideas are independent: 12 unit squares arranged as 1 by 12, 2 by 6 or 3 by 4 always have area 12 but perimeters 26, 16 and 14 units, and rectangles of perimeter 20 cm (1 by 9, 4 by 6, 5 by 5) have areas 9, 24 and 25 square cm, the square enclosing the most. 1 square metre is 100 cm × 100 cm = 10,000 square cm.
Nets, views and maps
A net is a flat pattern that folds into a solid. A cube has 11 different nets, but a straight row of 6 squares or a block of 2 by 3 squares cannot fold into a cube (the Class 5 chapter Boxes and Sketches). On a standard dice the opposite faces add up to 7 (1 and 6, 2 and 5, 3 and 4). Spatial understanding also means seeing an object from the top, front and side (the Class 3 chapter Where to Look From) and reading maps with a scale: if 1 cm stands for 100 m, then 4.5 cm on the map is 450 m on the ground.
Key facts
- Cube and cuboid: 6 faces, 12 edges, 8 vertices; triangular prism 5, 9, 6; square pyramid 5, 8, 5; faces + vertices - edges = 2.
- Cylinder: 2 flat faces, 1 curved surface, 2 curved edges, no vertex; cone: 1 flat face, 1 curved surface, 1 vertex; sphere: only a curved surface.
- van Hiele: at the visual level a tilted square is called a diamond; at the analysis level shapes are described by their properties.
- Lines of symmetry: square 4, non-square rectangle 2, equilateral triangle 3, isosceles 1, scalene 0, regular hexagon 6, circle infinitely many.
- Turn symmetry: a square looks the same after each 1/4 turn, a rectangle after a 1/2 turn; H, N, S, X and Z after a 1/2 turn.
- Tangram: 7 pieces (5 triangles, 1 square, 1 parallelogram); two small triangles equal the square, the medium triangle or the parallelogram in area.
- Same area, different perimeter: 12 unit squares as 1 by 12, 2 by 6 or 3 by 4 give perimeters 26, 16 and 14.
- Same perimeter 20 cm: rectangles 1 by 9, 4 by 6 and 5 by 5 have areas 9, 24 and 25 square cm; the square encloses the most.
- A cube has 11 nets; opposite faces of a dice add up to 7; on a map where 1 cm = 100 m, 4.5 cm = 450 m.
Quick revision
Solids before plane shapes; face, edge, vertex; cube 6 faces, 12 edges, 8 vertices; triangular prism 5, 9, 6; square pyramid 5, 8, 5; F + V - E = 2; diameter = 2 × radius; right angle 90 degrees, acute less than 90, obtuse between 90 and 180; symmetry lines: square 4, non-square rectangle 2, equilateral triangle 3, circle infinite; tangram 7 pieces; perimeter = boundary, area = unit squares, they vary independently; a cube has 11 nets.