---
title: "Shapes and Solids"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 11
exercises: 10
source: https://one-course.com/books/math/1/en/chapter/11-shapes-and-solids
---

# Chapter 11 — Shapes 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](#def-g2-shapes-solids) 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.](https://one-course.com/images/onecourse/chapters/math-1/g2-shapes/fig-cb93a64c2d3b.svg)

*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:

1. if the two sides of the shape run exactly along the folded corner, the angle is right;
2. 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:

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

![A figure and its copy, made by counting squares between the corners.](https://one-course.com/images/onecourse/chapters/math-1/g2-shapes/fig-586e82552827.svg)

*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](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#ch-g5-solids).

![The first solids. Run a finger over a cube: you feel 6 flat faces and sharp edges; a ball has none.](https://one-course.com/images/onecourse/chapters/math-1/g2-shapes/fig-3558235145c8.svg)

*The first solids. Run a finger over a [cube](#def-g2-shapes-solids): you feel $6$ flat [faces](#def-g2-shapes-solids) 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 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](#def-g2-shapes-sort)” or “not a [polygon](#def-g2-shapes-sort)”: a square; a circle; a triangle; a hexagon.

**Solution of Exercise 11.2.**

[Polygons](#def-g2-shapes-sort): square, triangle, hexagon (straight sides). Not a [polygon](#def-g2-shapes-sort): 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 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 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 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 of Exercise 11.6.**

Die: [cube](#def-g2-shapes-solids). Can of soup: [cylinder](#def-g2-shapes-solids). Football: ball. Shoebox: box.

**Exercise 11.7 ★.**

How many [faces](#def-g2-shapes-solids) has a [cube](#def-g2-shapes-solids)? What shape is each [face](#def-g2-shapes-solids)? How many [faces](#def-g2-shapes-solids) has a box?

**Solution of Exercise 11.7.**

A [cube](#def-g2-shapes-solids) has $6$ [faces](#def-g2-shapes-solids), each a square. A box also has $6$ [faces](#def-g2-shapes-solids) (rectangles).

**Exercise 11.8 ★.**

Which solids can roll: the [cube](#def-g2-shapes-solids), the box, the ball, the [cylinder](#def-g2-shapes-solids)? Why?

**Solution of Exercise 11.8.**

The ball rolls (round all over) and the [cylinder](#def-g2-shapes-solids) rolls (on its curved side). The [cube](#def-g2-shapes-solids) and the box do not: they have only flat [faces](#def-g2-shapes-solids) and stop on them.

**Exercise 11.9 ★.**

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

**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 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).