Nets of prisms and pyramids

Measures and solids, grade 6 Lesson 5 of 16Learning

Nets of prisms and pyramids

A net

Cut a cardboard box along some of its edges and spread it out flat on the table. What you get is a net of the solid.

A net shows all the solid’s faces, joined to each other along their sides.
Folding the net back along the joined sides gives the solid again.
One solid has several different nets: it depends on which edges you cut along.

A net of a prism

This is a net of a triangular prism:

The three rectangles are the lateral surface. Folded, they go around the prism, and the free side of the first rectangle meets the free side of the last one.
The two triangles are the bases: one closes the prism at the top, the other at the bottom.
Two sides that meet when folded become one edge, so they are equally long. For example, a side of a triangle is as long as the side of the rectangle it is joined to.
The net of every prism has two bases, and one rectangle for each side of the base.

Exercise 1

Which of the shapes are nets of a triangular prism?
Imagine folding each shape (or cut it out of paper and check), and explain.

A net of a pyramid

This is a net of a square pyramid:

The square is the base.
The four triangles are the lateral surface. Fold each triangle up, and their free vertices all meet at one point – the apex of the pyramid.

And this is a net of a triangular pyramid: a large triangle divided into four triangles.

The middle triangle is the base, and the three triangles around it fold up and meet at the apex.
The net of every pyramid has one base, and one triangle for each side of the base.

Exercise 2

Which solid does each net fold into?
Write the name of each solid.

Exercise 3

Dana wants to build a pentagonal prism out of pieces of cardboard.
Which polygons does she have to cut out, and how many of each?

Counting edges in a net

A net lets you count the solid’s edges without seeing the solid: every edge of the solid is made of two sides of faces, which meet when the net is folded.

For example, in a net of a triangular prism:

The three rectangles have $\displaystyle {3\times4}={12}$ sides, and the two triangles $\displaystyle {2\times3}={6}$ sides. Together: $\displaystyle {12+6}={18}$ sides.
Every two sides are one edge, so the prism has $\displaystyle {18:2}={9}$ edges – as we counted in the last lesson.

Exercise 4

Calculate with the net of a hexagonal prism (net 1 in Exercise 2).

How many sides do all the polygons in the net have together?

/ – 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 a hexagonal prism 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 5

The base of a pyramid is a square whose side is 4 squares of the grid.
Draw a net of it.
You can copy the net onto paper, cut it out and build the pyramid.

Formulas: measures and solids, grade 6

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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
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{2}{3}\times\pi r^2\times2r$