---
title: "Shapes and Right Angles"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 18
exercises: 11
source: https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles
---

# Chapter 18 — Shapes and Right Angles

Squares, rectangles, triangles, circles: the shapes are old friends. This chapter looks at them with a geometer’s eye — what exactly makes a square a square? — and introduces the tool that decides: the set square and its [right angle](#def-g3-shapes-rightangle).

## 18.1 The right angle

**Definition 18.1 (Right angle).**

A *right angle* is the corner angle of a sheet of paper, or of the set square. Two lines making a right angle are *perpendicular*. In figures, a right angle is marked with a small square in the corner.

**Method 18.2 (Checking a right angle).**

1. Place the corner of the set square exactly on the corner to test;
2. slide one edge of the set square along one side of the angle;
3. if the other side of the angle runs exactly along the other edge, the angle is right — if it opens wider or narrower, it is not.

![Only the first angle is right — and only the little square mark, or the set square, can certify it: eyes alone are easily fooled.](https://one-course.com/images/onecourse/chapters/math-1/g3-shapes/fig-680ddb391312.svg)

*Only the first angle is right — and only the little square mark, or the set square, can certify it: eyes alone are easily fooled.*

## 18.2 The shapes and their rules

**Definition 18.3 (Square, rectangle, triangle).**

- A *rectangle* has $4$ sides and $4$ [right angles](#def-g3-shapes-rightangle) ; its opposite sides have the same length.
- A *square* is a rectangle whose $4$ sides all have the same length.
- A *triangle* has $3$ sides; a *right triangle* has one [right angle](#def-g3-shapes-rightangle) .
- A *circle* is drawn with the compass: all its points are at the same distance (the *radius* ) from the *center* .

![The shape gallery, with their marks: little squares for right angles, small ticks for equal sides.](https://one-course.com/images/onecourse/chapters/math-1/g3-shapes/fig-9b7081ec890e.svg)

*The shape gallery, with their marks: little squares for [right angles](#def-g3-shapes-rightangle), small ticks for equal sides.*

**Example 18.4 (Why marks matter).**

A shape drawn slightly tilted can *look* like a square without being one. The marks tell the truth: $4$ right-angle squares $+$ $4$ tick marks $=$ a real square. When you draw, put the marks; when you read a figure, trust only the marks and the given measures.

## 18.3 Drawing on grid paper

**Method 18.5 (Constructions on a grid).**

Grid paper offers free [right angles](#def-g3-shapes-rightangle): its lines are perpendicular.

1. To draw a rectangle $5$ by $3$ : choose a crossing point, count $5$ squares right, $3$ squares up, and close the shape along the grid lines;
2. to check equal lengths, count squares;
3. diagonal steps make tilted lines: going “ $1$ right, $1$ up” repeatedly draws a $45^\circ$ line.

![Two constructions on the grid: the grid lines guarantee the right angles.](https://one-course.com/images/onecourse/chapters/math-1/g3-shapes/fig-a8660205c0f0.svg)

*Two constructions on the grid: the grid lines guarantee the [right angles](#def-g3-shapes-rightangle).*

**Example 18.6 (A construction program).**

A figure can be described by instructions: “1. Draw a square $ABCD$ of side $4$ cm. 2. Draw the circle with center $A$ passing through $B$.” Following a program line by line, anyone can redraw the same figure — and writing one forces you to name every point. Try it with a partner: one writes, the other draws.

## 18.4 Exercises

**Exercise 18.1 ★.**

In the classroom, find (without measuring) three [right angles](#def-g3-shapes-rightangle) and one angle that is not right. How can you *check* each one?

**Solution of Exercise 18.1.**

Typical [right angles](#def-g3-shapes-rightangle): the corners of the board, of a sheet of paper, of the door. A non-right angle: the opening of a pair of scissors, the hands of the clock at 1 o’clock. Each is checked with the set square ([Method 18.2](#met-g3-shapes-check)) — never just by eye.

**Exercise 18.2 ★.**

Draw a rectangle of $6$ cm by $4$ cm with ruler and set square. Mark its [right angles](#def-g3-shapes-rightangle) and its equal sides.

**Solution of Exercise 18.2.**

Construction: one side of $6$ cm, [right angles](#def-g3-shapes-rightangle) at both ends with the set square, sides of $4$ cm, close the shape. Marks: $4$ little squares in the corners, one tick on each $6$ cm side, a double tick on each $4$ cm side.

**Exercise 18.3 ★.**

Draw a square of side $5$ cm. How many [right angles](#def-g3-shapes-rightangle) does it have? How many equal sides? Mark everything.

**Solution of Exercise 18.3.**

A square has $4$ [right angles](#def-g3-shapes-rightangle) and $4$ equal sides — all marked.

**Exercise 18.4 ★.**

On grid paper, draw: a $4 \times 4$ square; a $7 \times 2$ rectangle; a right triangle with legs of $3$ and $5$ squares.

**Solution of Exercise 18.4.**

Grid constructions; the grid guarantees the [right angles](#def-g3-shapes-rightangle), and counting squares guarantees the lengths.

**Exercise 18.5 ★.**

True or false? Explain. “Every square is a rectangle.” “Every rectangle is a square.” “A triangle can have two [right angles](#def-g3-shapes-rightangle).” (Try to draw one!)

**Solution of Exercise 18.5.**

“Every square is a rectangle”: *true* — it has $4$ [right angles](#def-g3-shapes-rightangle), which is all a rectangle asks.

“Every rectangle is a square”: *false* — a $6 \times 4$ rectangle has unequal sides.

“A triangle can have two [right angles](#def-g3-shapes-rightangle)”: *false* — the two sides drawn from the two [right angles](#def-g3-shapes-rightangle) would be perpendicular to the same base line, hence never meet to close the triangle.

**Exercise 18.6 ★.**

Draw a circle of [radius](#def-g3-shapes-shapes) $4$ cm. Mark its center $O$, a point $A$ on the circle, and draw the [radius](#def-g3-shapes-shapes) $[OA]$. How long is $OA$? And if $B$ is another point of the circle, how long is $OB$?

**Solution of Exercise 18.6.**

$OA = 4$ cm (the [radius](#def-g3-shapes-shapes)). $OB = 4$ cm too: *every* point of the circle is at the same distance from the center.

**Exercise 18.7 ★.**

Follow this program: “1. Draw a rectangle $ABCD$ with $AB = 6$ cm and $BC = 3$ cm. 2. Mark the point $M$ in the middle of $[AB]$. 3. Join $M$ to $C$ and to $D$.” What shapes has the rectangle been cut into?

**Solution of Exercise 18.7.**

The rectangle is cut into three triangles: two right triangles ($AMD$ and $MBC$, right-angled at $A$ and $B$) and the middle triangle $MCD$.

**Exercise 18.8 ★.**

Sort the shapes: which of these have at least one [right angle](#def-g3-shapes-rightangle)? A square; a (non-square) rectangle; a right triangle; a circle; the letter L drawn with thick strokes.

**Solution of Exercise 18.8.**

With at least one [right angle](#def-g3-shapes-rightangle): the square, the rectangle, the right triangle, and the letter L (its inner and outer corners). The circle has no corner at all.

**Exercise 18.9 ★★.**

On grid paper, draw a tilted square whose sides go “$2$ right, $1$ up” (and the matching turns). Count squares to convince yourself the four sides have the same length.

**Solution of Exercise 18.9.**

Each side crosses “$2$ right, $1$ up” (or the turned versions “$1$ right, $2$ down”, etc.): every side is the diagonal of a $2 \times 1$ block of squares, so all four have the same length.

**Exercise 18.10 ★★.**

Write a construction program (numbered instructions, named points) for: a square of side $3$ cm sitting on top of a rectangle $3$ cm $\times$ $2$ cm, like a house on its ground floor. Test your program on a classmate.

**Solution of Exercise 18.10.**

One correct program: “1. Draw a rectangle $ABCD$ with $AB = 3$ cm (bottom) and $BC = 2$ cm. 2. On top of the side $[DC]$, draw the square $DCEF$ of side $3$ cm, outside the rectangle.” Any clear, numbered, complete program is correct — test it by following it word for word.

**Exercise 18.11 ★★.**

How many squares can you find in a $3 \times 3$ grid of small squares? (Count the $1 \times 1$, the $2 \times 2$ and the $3 \times 3$ ones.)

**Solution of Exercise 18.11.**

$1 \times 1$ squares: $9$. $2 \times 2$ squares: $4$. $3 \times 3$: $1$. Total: $9 + 4 + 1 = 14$ squares.