---
title: "Shapes and Space"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 5
exercises: 9
source: https://one-course.com/books/math/1/en/chapter/5-shapes-and-space
---

# Chapter 5 — Shapes and Space

Windows are rectangles, wheels are circles, roofs hide triangles: shapes are everywhere. This chapter learns to recognize them by counting sides and corners, and to say precisely *where* things are: left, right, above, below, between.

## 5.1 The four basic shapes

**Definition 5.1 (Square, rectangle, triangle, circle).**

- the *triangle* has $3$ sides and $3$ corners;
- the *square* has $4$ equal sides and $4$ corners;
- the *rectangle* has $4$ sides and $4$ corners, with opposite sides equal (a stretched square);
- the *circle* is perfectly round: no sides, no corners.

![The four basic shapes. To name a shape, do not trust the first glance: count its sides and corners.](https://one-course.com/images/onecourse/chapters/math-1/g1-shapes/fig-4e31d91a4a94.svg)

*The four basic shapes. To name a shape, do not trust the first glance: *count* its sides and corners.*

**Example 5.2 (Tilted shapes are still shapes).**

A square standing on its corner is still a square — four equal sides, four corners — even if it looks like a kite. Turning a shape does not change what it is.

**Example 5.3 (Shape hunt).**

In the classroom: the board is a rectangle, the clock a circle, a folded napkin can be a triangle, some windows are squares. Real objects are not perfect shapes, but their faces are close to them (solids and their faces return in [Chapter 11](https://one-course.com/books/math/1/en/chapter/11-shapes-and-solids#ch-g2-shapes)).

## 5.2 Where is it?

**Definition 5.4 (Position words).**

To say where something is: *left* / *right*, *above* / *below*, *on* / *under*, *in front of* / *behind*, *between*, *inside* / *outside*.

![Position words in action. Careful: your left and the left of someone facing you are opposite!](https://one-course.com/images/onecourse/chapters/math-1/g1-shapes/fig-fef321ececf8.svg)

*Position words in action. Careful: *your* left and the left of someone facing you are opposite!*

## 5.3 Paths on a grid

**Method 5.5 (Describing a path).**

On grid paper, a path is a list of steps: $\to$ (one square right), $\leftarrow$ (left), $\uparrow$ (up), $\downarrow$ (down).

1. To follow a path: start on the marked square and obey the arrows one by one;
2. to describe a path: walk it in your head and write the arrows in order;
3. two different paths can join the same two squares — some shorter, some longer.

![A path from start to goal, one arrow per step: \ \ \ \ \ \ — seven steps. Can you find another path with the same number of steps?](https://one-course.com/images/onecourse/chapters/math-1/g1-shapes/fig-8bbc18d487d8.svg)

*A path from start to goal, one arrow per step: $\to\ \to\ \uparrow\ \uparrow\ \to\ \to\ \to$ — seven steps. Can you find another path with the same number of steps?*

## 5.4 Exercises

**Exercise 5.1 ★.**

For each shape, count the sides and corners and name it: a shape with $3$ corners; a round shape with no corner; a shape with $4$ equal sides.

**Solution of Exercise 5.1.**

$3$ corners: a triangle. Round, no corner: a circle. $4$ equal sides: a square.

**Exercise 5.2 ★.**

Find at home: two rectangles, one circle, one triangle, one square. (Doors? plates? road signs?)

**Solution of Exercise 5.2.**

Answers vary: a door and a book are rectangles; a plate or a clock is a circle; a road-warning sign is a triangle; a tile can be a square.

**Exercise 5.3 ★.**

True or false? “A square turned on its corner stops being a square.” Explain with [Example 5.2](#ex-g1-shapes-tilted).

**Solution of Exercise 5.3.**

False: a square turned on its corner still has four equal sides and four corners — turning does not change the shape.

**Exercise 5.4 ★.**

Draw on grid paper: a square of $3$ squares by $3$; a rectangle of $5$ by $2$; a triangle using a diagonal of the grid.

**Solution of Exercise 5.4.**

Grid drawings: the square is $3 \times 3$ squares, the rectangle $5 \times 2$, and the triangle uses a slanted grid line as one side.

**Exercise 5.5 ★.**

Place three toys in a row. Describe the row with position words: which is between the others? Which is on the left?

**Solution of Exercise 5.5.**

Answers vary. Example: the ball is on the left, the car is between the ball and the doll, the doll is on the right.

**Exercise 5.6 ★.**

Draw a circle. Draw a cross *inside* it, a star *outside* it, and a dot *on* the circle itself.

**Solution of Exercise 5.6.**

The cross is drawn inside the circle, the star outside, and the dot right on the circle’s line.

**Exercise 5.7 ★.**

On grid paper, mark a start square. Follow: $\to\ \uparrow\ \to\ \uparrow\ \to$. Mark the arrival. Then write the path that goes straight back to the start.

**Solution of Exercise 5.7.**

Following $\to\ \uparrow\ \to\ \uparrow\ \to$ moves $3$ squares right and $2$ up. The way back is $\downarrow\ \downarrow$ then $\leftarrow\ \leftarrow\ \leftarrow$ (or any path returning to the start).

**Exercise 5.8 ★.**

How many corners in total: two triangles and one square? One circle and two rectangles?

**Solution of Exercise 5.8.**

Two triangles and one square: $3 + 3 + 4 = 10$ corners. One circle and two rectangles: $0 + 4 + 4 = 8$ corners.

**Exercise 5.9 ★★.**

Take $12$ matches. Can you outline a square using *all* of them, with whole matches and the same number on each side? How many matches per side? Could you do it with $10$ matches?

**Solution of Exercise 5.9.**

$12$ matches make a square with $3$ matches on each side ($3 + 3 + 3 + 3 = 12$). With $10$ matches it cannot be done: $10$ does not split into $4$ equal whole parts (each side would need $2$ matches and a half).