Pariksha SaathiLesson 6 of 8 · Mathematics, Classes VI-VIII (Paper II)
Geometry: Quadrilaterals, Circles, Symmetry, Constructions and Solid Shapes
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Polygons
A polygon is a simple closed curve made only of line segments; it is regular when all its sides and all its angles are equal. The angle sum of a polygon of n sides is (n − 2) × 180°, because the diagonals from one vertex cut it into n − 2 triangles: 360° for a quadrilateral, 540° for a pentagon, 720° for a hexagon. The exterior angles of a convex polygon, one at each vertex, add up to 360°, so each exterior angle of a regular polygon of n sides is 360°/n. A regular polygon with exterior angle 45° has 360 ÷ 45 = 8 sides, and one with interior angle 108° has exterior angle 72° and so 5 sides.
Quadrilaterals
A trapezium has a pair of parallel sides; a kite has two pairs of equal adjacent sides and perpendicular diagonals; a parallelogram has both pairs of opposite sides parallel. In a parallelogram opposite sides are equal, opposite angles are equal, adjacent angles are supplementary and the diagonals bisect each other, so one angle of 70° gives angles of 70°, 110°, 70° and 110°, and adjacent angles in the ratio 3 : 2 are 108° and 72°. A rhombus is a parallelogram with all sides equal, and its diagonals are perpendicular bisectors of each other; a rectangle is a parallelogram with every angle 90° and equal diagonals; a square is both. Every square is a rectangle and a rhombus, and every rectangle and rhombus is a parallelogram, but a rhombus whose angles are not 90° is not a rectangle.
Circles
A circle is the set of points at a fixed distance, the radius, from a fixed point, the centre. A chord joins two points of the circle; the diameter is a chord through the centre, the longest chord, equal to 2 × radius, and it divides the circle into two semicircles. An arc is a part of the circle, a sector is the region between an arc and two radii, and a segment is the region between an arc and a chord. The ratio of circumference to diameter is the same for every circle and is called π, approximately 22/7 or 3.14.
Symmetry
A figure has line symmetry if a fold along a line, the mirror line, makes its two halves coincide. An isosceles triangle has 1 line of symmetry, an equilateral triangle 3, a rectangle 2, a rhombus 2, a square 4, a regular polygon of n sides n, a scalene triangle none and a circle infinitely many. A figure has rotational symmetry if it looks exactly the same more than once in a full turn; the number of times is the order, and 360° divided by the order is the angle of rotation. A square has order 4 (90°), an equilateral triangle 3 (120°), a rectangle 2 (180°) and a regular hexagon 6 (60°). Some figures have both kinds (a square, the letter H), some only line symmetry (the letter A) and some only rotational symmetry (the letters S, N and Z, or a parallelogram that is neither a rectangle nor a rhombus).
Constructions
With a ruler and compasses one can copy a line segment, draw the perpendicular bisector of a segment, draw a perpendicular to a line from a point, bisect an angle, and make angles of 60° (from an equilateral triangle), 120° (two 60° steps), 90°, 30° (half of 60°), 45° (half of 90°), 15° and 75°. A triangle can be constructed from SSS, SAS, ASA or RHS data, the same as the congruence criteria, because each set fixes exactly one triangle; it cannot be constructed when the sides break the triangle inequality or two given angles add up to 180° or more.
Visualising solid shapes
A polyhedron is a solid with flat polygonal faces, straight edges and vertices; cylinders, cones and spheres have curved surfaces and are not polyhedra. Every convex polyhedron obeys Euler's formula F + V − E = 2: a cube or cuboid has 6 faces, 8 vertices and 12 edges; a triangular prism 5, 6 and 9; a square pyramid 5, 5 and 8; a tetrahedron 4, 4 and 6; a pentagonal prism 7, 10 and 15. So a polyhedron with 12 vertices and 30 edges has 20 faces, and none can have 10 faces, 20 edges and 15 vertices, since 10 + 15 − 20 = 5. A net is a flat shape that folds into a solid (a cube has 11 different nets), and opposite faces of a die add up to 7.
In the classroom
Pupils who know a square only in its usual position may deny that a tilted square is a square, or think a rhombus can never be a square. Showing shapes in many orientations, sorting quadrilaterals by their properties, folding cut-outs to find lines of symmetry, and building solids from straws and clay to count F, V and E move pupils from the van Hiele visualisation level, where shapes are known by their look, to the analysis level, where they are known by their properties.
Key facts
- Angle sum of a polygon of n sides = (n − 2) × 180°: pentagon 540°, hexagon 720°; the exterior angles of a convex polygon add up to 360°.
- Regular polygons: exterior angle 45° means 8 sides; interior angle 108° means exterior angle 72° and 5 sides; a regular hexagon has interior angles of 120°.
- Parallelogram: one angle of 70° gives 70°, 110°, 70°, 110°; adjacent angles in the ratio 3 : 2 are 108° and 72°.
- Diagonals bisect each other in every parallelogram, are equal in a rectangle, perpendicular in a rhombus and a kite, and equal and perpendicular in a square.
- Circle: diameter = 2 × radius and is the longest chord; a sector is bounded by two radii and an arc, a segment by a chord and an arc.
- Lines of symmetry: isosceles triangle 1, equilateral triangle 3, rectangle 2, square 4, regular hexagon 6, circle infinitely many.
- Rotational symmetry: square order 4 (90°), equilateral triangle order 3 (120°), regular hexagon order 6 (60°); S and Z have order 2 but no line of symmetry.
- Ruler-and-compasses angles: 60°, 120°, 90°, 30°, 45°, 15°, 75°; triangles are constructed from SSS, SAS, ASA or RHS data.
- Euler's formula F + V − E = 2: cube 6 + 8 − 12 = 2; triangular prism 5 + 6 − 9 = 2; square pyramid 5 + 5 − 8 = 2.
- A cube has 11 different nets, and the opposite faces of a die add up to 7.
Quick revision
Polygon angle sum (n − 2) × 180°, exterior angles 360°, regular polygon exterior angle 360°/n; parallelogram: opposite sides and angles equal, adjacent angles add up to 180°, diagonals bisect; rhombus diagonals perpendicular, rectangle diagonals equal, square both; diameter = 2r, longest chord; lines of symmetry: square 4, equilateral triangle 3, rectangle 2, circle infinite; rotation angle = 360°/order; ruler-compass angles 60°, 120°, 90°, 30°, 45°; Euler F + V − E = 2.