Primary & Middle School Mathematics · Grades 1–9

11Shapes and Solids

Flat shapes live on paper; solids fill your hands. This chapter sorts flat shapes by counting sides and corners, tests the right angle with a folded paper corner, and meets the first solids — whose faces are the flat shapes we already know.

11.1 Sorting flat shapes

Definition 11.1 (Straight-sided and round)

A shape whose border is made only of straight sides is a polygon: triangle ( $3$ sides), quadrilateral ( $4$ sides: squares and rectangles are quadrilaterals), pentagon ( $5$ ), hexagon ( $6$ ). A shape with a curved border, like the circle, is not a polygon.

Sorting by the number of sides. The number of corners always equals the number of sides.

Method 11.2 (Testing a right angle)

Fold a sheet of paper twice to make a sharp corner — that corner is a right angle. Lay it into the corner of a shape:

if the two sides of the shape run exactly along the folded corner, the angle is right;
if the corner is wider or narrower, it is not right.

A square and a rectangle have four right angles.

11.2 Reproducing on a grid

Method 11.3 (Copying a figure)

On grid paper, copy a figure vertex by vertex:

pick a corner and mark it on the empty grid;
count the squares to the next corner (“ $3$ right, $2$ up”) and mark it;
join the corners in order; compare with the model.

A figure and its copy, made by counting squares between the corners.

11.3 Solids

Definition 11.4 (First solids)

Solids take up space. Their flat sides are faces:

the cube: $6$ square faces (a die);
the box (rectangular prism): $6$ rectangle faces (a cereal packet);
the cylinder: two round faces and a rolled side (a can);
the ball: perfectly round, no flat face.

Solids and their faces return in Chapter 35.

The first solids. Run a finger over a cube: you feel 6 flat faces and sharp edges; a ball has none. — The first solids. Run a finger over a cube: you feel $6$ flat faces and sharp edges; a ball has none.

11.4 Exercises

Exercise 11.1 ★

For each, count sides and corners and name it: a shape with $5$ sides; a shape with $3$ corners; a round shape.

Solution

Solution of Exercise 11.1.

$5$ sides, $5$ corners: a pentagon. $3$ corners (and $3$ sides): a triangle. Round: a circle.

Exercise 11.2 ★

Sort into “polygon” or “not a polygon”: a square; a circle; a triangle; a hexagon.

Solution

Solution of Exercise 11.2.

Polygons: square, triangle, hexagon (straight sides). Not a polygon: the circle (curved border).

Exercise 11.3 ★

Make a right-angle tester by folding paper. Find three right angles in the room, and one angle that is not right.

Solution

Solution of Exercise 11.3.

Right angles at, for instance, the corners of the board, a book, a window; a not-right angle at the open lid of a box or the hands of a clock at $2$ o’clock.

Exercise 11.4 ★

How many right angles has a square? A rectangle? A triangle can have at most how many? (Try to draw one with two.)

Solution

Solution of Exercise 11.4.

Square: $4$ right angles. Rectangle: $4$ . A triangle can have at most $1$ right angle — two would make the other two sides never meet.

Exercise 11.5 ★

On grid paper, copy a rectangle that is $4$ squares wide and $2$ tall, then a triangle with a $3$ -square base.

Solution

Solution of Exercise 11.5.

Grid copies made by counting squares between corners; comparing with the model checks the work.

Exercise 11.6 ★

Name the solid: a die; a can of soup; a football; a shoebox.

Solution

Solution of Exercise 11.6.

Die: cube. Can of soup: cylinder. Football: ball. Shoebox: box.

Exercise 11.7 ★

How many faces has a cube? What shape is each face? How many faces has a box?

Solution

Solution of Exercise 11.7.

A cube has $6$ faces, each a square. A box also has $6$ faces (rectangles).

Exercise 11.8 ★

Which solids can roll: the cube, the box, the ball, the cylinder? Why?

Solution

Solution of Exercise 11.8.

The ball rolls (round all over) and the cylinder rolls (on its curved side). The cube and the box do not: they have only flat faces and stop on them.

Exercise 11.9 ★

A shape has $6$ sides. What is it called? How many corners has it?

Solution

Solution of Exercise 11.9.

$6$ sides: a hexagon, with $6$ corners.

Exercise 11.10 ★★

On grid paper, draw a house: a square for the wall ( $4 \times 4$ ) with a triangle roof on top. How many sides and corners has the whole outline of the house?

Solution

Solution of Exercise 11.10.

The house outline (square wall with a triangle roof, the roof base sitting on the square’s top side) has $5$ sides and $5$ corners — it is a pentagon (bottom, two walls, two roof slopes; the top of the wall is hidden under the roof).