Important Formulas: Shapes and Patterns | Mathematics for Class 5 PDF Download

Patterns

• A pattern is a sequence that repeats in the same way.

• Patterns can be made with shapes, numbers, or colours.

• Rule of a pattern → helps predict the next item.

• Examples:

  • Colours: red, blue, red, blue → next is red.

  • Shapes: circle, square, circle, square → keeps repeating.

  • Numbers: 2, 4, 6, 8 → rule: add 2 each time.

Key Uses of Patterns:

1. Organise information.

2. Solve problems easily.

3. Predict what comes next.

4. Seen in nature (zebra stripes, honeycombs, petals), art, tiles, and weaving.

Weaving Mats

• Weaving rule: Alternate “over–under” rows to form a mat.

• Pattern created: Checkerboard / basket design.

• Different weaving orders = different patterns.

Shapes and Their Properties

Polygons

• A polygon is a closed shape made of straight sides.

Tessellation (Tiling Patterns)

Tessellation = covering a surface with shapes without gaps/overlaps.

Examples:

• Squares → tessellate.

• Equilateral triangles → tessellate (6 around a point).

• Hexagons → tessellate (3 around a point).

• Pentagons & octagons → do not tessellate (leave gaps).

Mixed Tessellation: Shapes like triangles + hexagons, or rhombuses + triangles can combine to tessellate.

Types of Triangles

1. Equilateral → 3 equal sides, 3 equal angles (60°).

2. Isosceles → 2 equal sides, opposite angles equal.

3. Scalene → All sides and angles different.

Quadrilaterals

• Parallelogram → Opposite sides equal, opposite angles equal.

• Rectangle → A parallelogram with all angles = 90°.

• Square → A rectangle with all sides equal.

• Rhombus → All sides equal, angles not 90°.

• Trapezium → One pair of parallel sides.

Tangram

A Tangram has 7 pieces (tans):

• 2 Large right-angled isosceles triangles

• 1 Medium right-angled isosceles triangle

• 2 Small right-angled isosceles triangles

• 1 Square

• 1 Parallelogram

Key Points:

• Can form many shapes (bird, fish, square, trapezium, etc.).

• Develops spatial thinking and creativity.

Kite

• 4-sided quadrilateral with two pairs of adjacent equal sides.

• Diagonals cross at right angles.

• Seen in flying kites, diamond shapes.

• Can be made from folding a square.

Circles

• Radius (r): Distance from centre to edge.

• Diameter (d): Longest distance across circle, d = 2r.

• Circumference (C): Distance around circle, C = 2πr.

• Area (A): Space inside circle, A = πr².

• Angles around circle: 360°.

Key Property:

• Joining endpoints of two diameters always makes a rectangle (or square if diameters are perpendicular).

3D Shapes

Cube

• 6 square faces, 12 edges, 8 vertices.

• Surface Area = 6 × side²

• Volume = side³

Icosahedron

• 20 equilateral triangle faces.

• 30 edges, 12 vertices.

Dodecahedron

• 12 regular pentagon faces.

• 30 edges, 20 vertices.

Both are Platonic solids (faces are identical regular polygons, same number of faces meet at each vertex)