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There are **45** NRICH Mathematical resources connected to **Polyhedra**, you may find related items under 3D geometry, shape and space.

Problem
Primary curriculum
Secondary curriculum
### Guess What?

Can you find out which 3D shape your partner has chosen before they work out your shape?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Which Solids Can We Make?

Interior angles can help us to work out which polygons will tessellate. Can we use similar ideas to predict which polygons combine to create semi-regular solids?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Next Size Up

The challenge for you is to make a string of six (or more!) graded cubes.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Octa-flower

Join some regular octahedra, face touching face and one vertex of each meeting at a point. How many octahedra can you fit around this point?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Shadow Play

Here are shadows of some 3D shapes. What shapes could have made them?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Cut Nets

Each of the nets of nine solid shapes has been cut into two pieces. Can you see which pieces go together?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Tetra Perp

Show that the edges $AD$ and $BC$ of a tetrahedron $ABCD$ are mutually perpendicular if and only if $AB^2 +CD^2 = AC^2+BD^2$. This problem uses the scalar product of two vectors.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Pythagoras for a Tetrahedron

In a right-angled tetrahedron prove that the sum of the squares of the areas of the 3 faces in mutually perpendicular planes equals the square of the area of the sloping face. A generalisation of Pythagoras' Theorem.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Triangles to Tetrahedra

Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Skeleton Shapes

How many balls of modelling clay and how many straws does it take to make these skeleton shapes?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Tet-trouble

Is it possible to have a tetrahedron whose six edges have lengths 10, 20, 30, 40, 50 and 60 units?

Age 14 to 16

Challenge Level

Article
Primary curriculum
Secondary curriculum
### Going Deeper: Achieving Greater Depth with Geometry

This article for Primary teachers outlines how providing opportunities to engage with increasingly complex problems, and to communicate thinking, can help learners 'go deeper' with geometry.

Age 5 to 11

Article
Primary curriculum
Secondary curriculum
### Let's Get Flexible with Geometry

In this article for primary teachers, Ems explores ways to develop mathematical flexibility through geometry.

Age 5 to 11

Problem
Primary curriculum
Secondary curriculum
### More Dicey Decisions

The twelve edge totals of a standard six-sided die are distributed symmetrically. Will the same symmetry emerge with a dodecahedral die?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Which Solid?

This task develops spatial reasoning skills. By framing and asking questions a member of the team has to find out what mathematical object they have chosen.

Age 7 to 16

Challenge Level

General
Primary curriculum
Secondary curriculum
### Modular Origami Polyhedra

These models have appeared around the Centre for Mathematical Sciences. Perhaps you would like to try to make some similar models of your own.

Age 7 to 16

Challenge Level

General
Primary curriculum
Secondary curriculum
### Platonic and Archimedean Solids

In a recent workshop, students made these solids. Can you think of reasons why I might have grouped the solids in the way I have before taking the pictures?

Age 7 to 16

Challenge Level

Article
Primary curriculum
Secondary curriculum
### Investigating Solids with Face-transitivity

In this article, we look at solids constructed using symmetries of their faces.

Age 14 to 18

Problem
Primary curriculum
Secondary curriculum
### Magnetic Personality

60 pieces and a challenge. What can you make and how many of the pieces can you use creating skeleton polyhedra?

Age 7 to 16

Challenge Level

Article
Primary curriculum
Secondary curriculum
### Paper Folding - Models of the Platonic Solids

A description of how to make the five Platonic solids out of paper.

Age 11 to 16

Problem
Primary curriculum
Secondary curriculum
### Sliced

An irregular tetrahedron has two opposite sides the same length a and the line joining their midpoints is perpendicular to these two edges and is of length b. What is the volume of the tetrahedron?

Age 14 to 16

Challenge Level

Article
Primary curriculum
Secondary curriculum
### Thinking 3D

How can we as teachers begin to introduce 3D ideas to young children? Where do they start? How can we lay the foundations for a later enthusiasm for working in three dimensions?

Age 7 to 14

Problem
Primary curriculum
Secondary curriculum
### Tetrahedra Tester

An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### A Chain of Eight Polyhedra

Can you arrange the shapes in a chain so that each one shares a face (or faces) that are the same shape as the one that follows it?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### The Dodecahedron

What are the shortest distances between the centres of opposite faces of a regular solid dodecahedron on the surface and through the middle of the dodecahedron?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Dodecawhat

Follow instructions to fold sheets of A4 paper into pentagons and assemble them to form a dodecahedron. Calculate the error in the angle of the not perfectly regular pentagons you make.

Age 14 to 16

Challenge Level

Article
Primary curriculum
Secondary curriculum
### Euler's Formula and Topology

Here is a proof of Euler's formula in the plane and on a sphere together with projects to explore cases of the formula for a polygon with holes, for the torus and other solids with holes and the relationship between Euler's formula and angle deficiency of polyhedra.

Age 16 to 18

Problem
Primary curriculum
Secondary curriculum
### Classifying Solids Using Angle Deficiency

Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry

Age 11 to 16

Challenge Level

Article
Primary curriculum
Secondary curriculum
### The Dodecahedron Explained

What is the shortest distance through the middle of a dodecahedron between the centres of two opposite faces?

Age 16 to 18

Problem
Primary curriculum
Secondary curriculum
### Icosian Game

This problem is about investigating whether it is possible to start at one vertex of a platonic solid and visit every other vertex once only returning to the vertex you started at.

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Child's Play

A toy has a regular tetrahedron, a cube and a base with triangular and square hollows. If you fit a shape into the correct hollow a bell rings. How many times does the bell ring in a complete game?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Rhombicubocts

Each of these solids is made up with 3 squares and a triangle around each vertex. Each has a total of 18 square faces and 8 faces that are equilateral triangles. How many faces, edges and vertices does each solid have?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Tetra Square

ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square.

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Dodecamagic

Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Redblue

Investigate the number of paths you can take from one vertex to another in these 3D shapes. Is it possible to take an odd number and an even number of paths to the same vertex?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### A Mean Tetrahedron

Can you number the vertices, edges and faces of a tetrahedron so that the number on each edge is the mean of the numbers on the adjacent vertices and the mean of the numbers on the adjacent faces?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Face Painting

You want to make each of the 5 Platonic solids and colour the faces so that, in every case, no two faces which meet along an edge have the same colour.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Tetrahedron Faces

One face of a regular tetrahedron is painted blue and each of the remaining faces are painted using one of the colours red, green or yellow. How many different possibilities are there?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Tetra Inequalities

Can you prove that in every tetrahedron there is a vertex where the three edges meeting at that vertex have lengths which could be the sides of a triangle?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Reach for Polydron

A tetrahedron has two identical equilateral triangles faces, of side length 1 unit. The other two faces are right angled isosceles triangles. Find the exact volume of the tetrahedron.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Platonic Planet

Glarsynost lives on a planet whose shape is that of a perfect regular dodecahedron. Can you describe the shortest journey she can make to ensure that she will see every part of the planet?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Proximity

We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.

Age 14 to 16

Challenge Level