Resources tagged with: Polyhedra

Filter by: Content type:
Age range:
Challenge level:

There are 42 NRICH Mathematical resources connected to Polyhedra, you may find related items under 3D Geometry, Shape and Space.

Broad Topics > 3D Geometry, Shape and Space > Polyhedra

problem icon

Next Size Up

Age 7 to 11 Challenge Level:

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

problem icon

Octa-flower

Age 16 to 18 Challenge Level:

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?

problem icon

Shadow Play

Age 5 to 7 Challenge Level:

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

problem icon

Cut Nets

Age 7 to 11 Challenge Level:

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

problem icon

Tetra Perp

Age 16 to 18 Challenge Level:

Show that the edges AD and BC of a tetrahedron ABCD are mutually perpendicular when: AB²+CD² = AC²+BD².

problem icon

Pythagoras for a Tetrahedron

Age 16 to 18 Challenge Level:

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. . . .

problem icon

Triangles to Tetrahedra

Age 11 to 14 Challenge Level:

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

problem icon

Skeleton Shapes

Age 5 to 7 Challenge Level:

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

problem icon

Tet-trouble

Age 14 to 16 Challenge Level:

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

problem icon

Three Cubes

Age 14 to 16 Challenge Level:

Can you work out the dimensions of the three cubes?

problem icon

More Dicey Decisions

Age 16 to 18 Challenge Level:

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

problem icon

Which Solids Can We Make?

Age 11 to 14 Challenge Level:

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?

problem icon

Which Solid?

Age 7 to 16 Challenge Level:

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.

problem icon

Plaited Origami Polyhedra

Age 7 to 16 Challenge Level:

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

problem icon

Platonic and Archimedean Solids

Age 7 to 16 Challenge Level:

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?

problem icon

Investigating Solids with Face-transitivity

Age 14 to 18

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

problem icon

Lighting up Time

Age 7 to 14 Challenge Level:

A very mathematical light - what can you see?

problem icon

Magnetic Personality

Age 7 to 16 Challenge Level:

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

problem icon

Paper Folding - Models of the Platonic Solids

Age 11 to 16

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

problem icon

Making Maths: Triangular Ball

Age 7 to 11 Challenge Level:

Make a ball from triangles!

problem icon

Sliced

Age 14 to 16 Challenge Level:

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?

problem icon

Thinking 3D

Age 7 to 14

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?

problem icon

Tetrahedra Tester

Age 11 to 14 Challenge Level:

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?

problem icon

A Chain of Eight Polyhedra

Age 7 to 11 Challenge Level:

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?

problem icon

The Dodecahedron

Age 16 to 18 Challenge Level:

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?

problem icon

Dodecawhat

Age 14 to 16 Challenge Level:

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.

problem icon

Euler's Formula and Topology

Age 16 to 18

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. . . .

problem icon

Classifying Solids Using Angle Deficiency

Age 11 to 16 Challenge Level:

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

problem icon

The Dodecahedron Explained

Age 16 to 18

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

problem icon

Icosian Game

Age 11 to 14 Challenge Level:

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.

problem icon

Child's Play

Age 7 to 11 Challenge Level:

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?

problem icon

Rhombicubocts

Age 11 to 14 Challenge Level:

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. . . .

problem icon

Tetra Square

Age 11 to 14 Challenge Level:

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.

problem icon

Dodecamagic

Age 7 to 11 Challenge Level:

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?

problem icon

Redblue

Age 7 to 11 Challenge Level:

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?

problem icon

A Mean Tetrahedron

Age 11 to 14 Challenge Level:

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?

problem icon

Face Painting

Age 7 to 11 Challenge Level:

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.

problem icon

Tetrahedron Faces

Age 7 to 11 Challenge Level:

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?

problem icon

Tetra Inequalities

Age 16 to 18 Challenge Level:

Prove that in every tetrahedron there is a vertex such that the three edges meeting there have lengths which could be the sides of a triangle.

problem icon

Reach for Polydron

Age 16 to 18 Challenge Level:

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.

problem icon

Platonic Planet

Age 14 to 16 Challenge Level:

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?

problem icon

Proximity

Age 14 to 16 Challenge Level:

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