Primary & Middle School Mathematics · Grades 1–9

18Shapes 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.

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)

Place the corner of the set square exactly on the corner to test;
slide one edge of the set square along one side of the angle;
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.

18.2 The shapes and their rules

Definition 18.3 (Square, rectangle, triangle)

A rectangle has $4$ sides and $4$ right angles; 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.
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.

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: its lines are perpendicular.

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;
to check equal lengths, count squares;
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.

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 and one angle that is not right. How can you check each one?

Solution

Solution of Exercise 18.1.

Typical right angles: 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) — 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 and its equal sides.

Solution

Solution of Exercise 18.2.

Construction: one side of $6$ cm, right angles 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 does it have? How many equal sides? Mark everything.

Solution

Solution of Exercise 18.3.

A square has $4$ right angles 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

Solution of Exercise 18.4.

Grid constructions; the grid guarantees the right angles, 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.” (Try to draw one!)

Solution

Solution of Exercise 18.5.

“Every square is a rectangle”: true — it has $4$ right angles, 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”: false — the two sides drawn from the two right angles would be perpendicular to the same base line, hence never meet to close the triangle.

Exercise 18.6 ★

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

Solution

Solution of Exercise 18.6.

$OA = 4$ cm (the radius). $OB = 4$ 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

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? A square; a (non-square) rectangle; a right triangle; a circle; the letter L drawn with thick strokes.

Solution

Solution of Exercise 18.8.

With at least one right angle: 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

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

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

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.