The prism and the pyramid

Measures and solids, grade 6 Lesson 4 of 16Learning

The prism and the pyramid

Solids

A solid is a shape in space: it has a length, a width and a height, and it takes up room. A shoebox, a drink can, a football and an ice-cream cone are solids.

Some solids have only flat surfaces, like a shoebox.
Others have a curved surface, like a can or a ball.
The prism and the pyramid are two kinds of solids whose surfaces are all flat.

Faces, edges and vertices

A solid whose surfaces are all flat has three kinds of parts:

Face – a flat surface of the solid. Every face is a polygon. In the drawing, one face is coloured.
Edge – a segment where two faces meet. In the drawing, one edge is thick.
Vertex (plural vertices) – a point where several edges meet. In the drawing, one vertex is marked.
The dashed lines are edges at the back: you cannot see them from here, but they are there.

The prism

A prism is a solid like a shoebox, a triangular chocolate box or a hexagonal pencil:

The bases of the prism – two congruent polygons (exactly the same), one above the other. In the drawing they are coloured.
The side faces are rectangles: each rectangle joins a side of the bottom base to the side above it in the top base. All the side faces together are the prism’s lateral surface.
A prism is named after its base: 1 is a triangular prism, 3 is a hexagonal prism, and there are square, pentagonal prisms and more.
A prism whose base is a rectangle is a box (2). A cube is a box whose edges are all equal.
A prism lying on its side is still a prism: its bases are the two congruent faces opposite each other, and all its other faces are rectangles.

The pyramid

A pyramid is a solid like the pyramids of Egypt or the roof of a tower:

The base of the pyramid – one polygon. In the drawing it is coloured.
The apex of the pyramid – a vertex above the base.
The side faces are triangles: each triangle joins a side of the base to the apex. Together they are the pyramid’s lateral surface.
A pyramid is named after its base too: 1 is a triangular pyramid, 2 a square pyramid, 3 a pentagonal pyramid. The pyramids of Egypt are square pyramids.

Exercise 1

Here are four solids.
For each one, write whether it is a prism or a pyramid, and what its base is.

Counting faces, edges and vertices

You can count a solid’s parts one by one, but it is easy to slip and count a part twice. It is safer to count by the base.

A triangular prism – its base has three sides:

Faces: two bases, and one side face for each side of the base: $\displaystyle {2+3}={5}$ .
Edges: three in the bottom base, three in the top base, and three standing between them: $\displaystyle {3+3+3}={9}$ .
Vertices: three at the bottom and three at the top: $\displaystyle {3+3}={6}$ .

A square pyramid – its base has four sides:

Faces: the base, and one triangle for each side of the base: $\displaystyle {1+4}={5}$ .
Edges: four in the base, and four going up from the base to the apex: $\displaystyle {4+4}={8}$ .
Vertices: four in the base, and the apex: $\displaystyle {4+1}={5}$ .

Solid	Faces	Edges	Vertices
Triangular prism	5	9	6
Square pyramid	5	8	5

Exercise 2

The bases of a hexagonal prism are hexagons.
Answer by calculating.

How many faces does it have?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

How many edges does it have?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

How many vertices does it have?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

The base of a pentagonal pyramid is a pentagon.
Answer by calculating.

How many faces does it have?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

How many edges does it have?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

How many vertices does it have?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Counting by the number of the base’s sides

In every prism and every pyramid, everything follows from the number of sides of the base:

Each side of the base gives one side face.
Each vertex of the base gives one edge going up from it: in a prism to the top base, in a pyramid to the apex.

So when the base has, say, eight sides:

	Faces	Edges	Vertices
Prism	$\displaystyle {8+2}={10}$	$\displaystyle {8\times3}={24}$	$\displaystyle {8\times2}={16}$
Pyramid	$\displaystyle {8+1}={9}$	$\displaystyle {8\times2}={16}$	$\displaystyle {8+1}={9}$

Exercise 4

A prism has 15 edges.
Which polygon is its base?
How many faces and how many vertices does it have?

Faces

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Vertices

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 5

A solid has seven faces and seven vertices.
Is it a prism or a pyramid? What is its base?
Explain.

Formulas: measures and solids, grade 6

Open as a page

The circle and the disc

The diameter $\displaystyle d$ is two radii $\displaystyle r$
$\displaystyle d$ $\displaystyle =$ $\displaystyle 2\times r$
The number pi – the value to calculate with
$\displaystyle \pi\approx3.14$
The circumference of a circle – $\displaystyle d$ the diameter
$\displaystyle C$ $\displaystyle =$ $\displaystyle \pi d$
The circumference of a circle – $\displaystyle r$ the radius
$\displaystyle C$ $\displaystyle =$ $\displaystyle 2\pi r$
The area of a disc – $\displaystyle r$ the radius
$\displaystyle A$ $\displaystyle =$ $\displaystyle \pi r^2$

Volumes

The volume of a box – $\displaystyle a$ the length, $\displaystyle b$ the width, $\displaystyle c$ the height
$\displaystyle V$ $\displaystyle =$ $\displaystyle a\times b\times c$
The volume of a box, a prism or a cylinder – $\displaystyle S$ the area of the base, $\displaystyle h$ the height
$\displaystyle V$ $\displaystyle =$ $\displaystyle S\times h$
The volume of a cylinder – $\displaystyle r$ the radius of the base, $\displaystyle h$ the height
$\displaystyle V$ $\displaystyle =$ $\displaystyle \pi r^2 h$
The volume of a pyramid – a third of the prism with the same base and height
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{S\times h}{3}$
The volume of a cone – a third of the cylinder with the same base and height
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{\pi r^2 h}{3}$
The volume of a sphere – two thirds of the cylinder around it
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{2}{3}\times\pi r^2\times2r$