7. Shapes and Patterns

Row 1 — 2 over, 1 under, 2 over, 1 under, … (repeat)

Row 2 — 2 under, 1 over, 2 under, 1 over, … (repeat)

Repeat Row 1 and Row 2 one after the other.

No. The gap left is smaller than a pentagon. So, regular pentagons do not tessellate.

Yes. 6 equilateral triangles fit together around a point without any gap or overlap.

Yes. 4 squares fit around a point.

No. 4 squares already fill the space around a point completely. There is no space for a fifth square.

Yes. 3 regular hexagons fit together at a point.

Regular hexagons and equilateral triangles.

No. Regular octagons leave square-shaped gaps. So, they do not tessellate on their own.

Two regular octagons and one square meet at each marked point.

Yes, the same set of shapes meets at every marked point.

No. Each triangle has only two equal sides. The third side is shorter. So, they are not equilateral.

Two squares and three triangles meet at each marked point.

Yes, the same set of shapes meets at each marked point.

Two triangles of the rhombus can make: a bigger triangle (two types), a rectangle, a parallelogram and a kite.

We can make 2 different triangles — one tall and one wide.

Each triangle has 2 equal sides. These are isosceles triangles.

On folding in half, the two angles at the base match. So, an isosceles triangle has 2 equal angles.

No. With these pieces we cannot make a triangle with all three sides equal.

Yes. Each single piece of the rhombus is a triangle with all three sides unequal.

Yes, all three angles of an equilateral triangle are equal.

After cutting it in half, each new triangle has no equal sides. It is a scalene triangle.

In a scalene triangle, no two angles are equal.

We can make a kite, a parallelogram and a rectangle.

In a kite, two pairs of adjacent sides are equal. Side 1 = Side 2 and Side 3 = Side 4.

In both A and B, the opposite sides are equal. So, both are parallelograms.

A has 2 acute angles and 2 obtuse angles. Its opposite angles are equal.

B has 4 right angles. All its angles are equal. So, B is a rectangle.

(a) Square — No. The sides of the pieces are not equal.

(b) Rectangle — Yes. Make two rectangles with 2 pieces each and join them.

(c) Triangle — Yes. Three pieces in the same direction and one turned upside down in the middle make a big triangle.

(d) Pentagon — Yes (like a house shape with one slanted side).

(e) Hexagon — Yes. Take the big triangle and move one corner piece to another side.

(f) Octagon — No.

The 7 pieces are: 2 big triangles, 1 medium triangle, 2 small triangles, 1 square and 1 parallelogram.

(a) All the triangles have the same shape but different sizes. The square and the parallelogram are 4-sided.

(b) Each triangle has one right angle and two equal angles. The square has 4 right angles. The parallelogram has 2 acute and 2 obtuse angles.

(c) Each triangle has two equal sides. The square has all sides equal. The parallelogram has opposite sides equal.

Yes. Statements 3, 4, 6 and 7 describe more than one shape.

We see triangles in the kite. The crease divides the kite into two equal triangles, and the folded flaps are also triangles.

A rectangle is formed. Its opposite sides are equal and all its angles are right angles.

With a different pair of diameters, we again get a rectangle.

If the two diameters are at right angles to each other, we get a square (a special rectangle).

No. This process always gives a rectangle.

Purple (with the green diamond) and brown are opposite faces.

Blue (circle), yellow (cross) and pink (star) come one after the other around the cube.

On the net: put yellow in the middle, purple above it and brown below it. Put blue on its left and pink on its right. The last square (opposite yellow) is the face with a cross seen next to blue in the third view.

(a) 3 × 3 × 3 = 27 cubes. The frame has 20 cubes. Removed = 27 − 20 = 7 cubes

(b) 4 × 4 × 4 = 64 cubes. The frame has 32 cubes. Removed = 64 − 32 = 32 cubes

(c) 5 × 5 × 5 = 125 cubes. The frame has 44 cubes. Removed = 125 − 44 = 81 cubes

(a) three faces painted red? 8 (the corner cubes)

(b) two faces painted red? 12 (the middle cube of each edge)

(c) one face painted red? 6 (the centre cube of each face)

(d) no faces painted red? 1 (the cube in the very centre)

Circle, Square, Rectangle, Triangle, Square, Triangle, Hexagon

Check: The first square is between the circle and the rectangle. The rectangle is between a square and a triangle. The two triangles are next to the second square. The hexagon is to the right of the triangle. The circle is to the left of the square.

Name: “Icosa” means 20 and “dodeca” means 12. “Hedron” means faces.

Counting edges: Each face of the icosahedron has 3 edges: 20 × 3 = 60. Each edge is shared by 2 faces, so 60 ÷ 2 = 30. For the dodecahedron: 12 × 5 = 60, and 60 ÷ 2 = 30.

Other solids with faces that look the same: cube (3 squares at each vertex), tetrahedron (3 triangles at each vertex) and octahedron (4 triangles at each vertex). Yes, the same number of faces meets at each vertex.