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

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Tetra Perp

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Pythagoras for a Tetrahedron

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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. . . .

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Triangles to Tetrahedra

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Starting with four different triangles, imagine you have an unlimited number of each type. How many different tetrahedra can you make? Convince us you have found them all.

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Tet-trouble

Stage: 4 Challenge Level: Challenge Level:1

Show that is it impossible to have a tetrahedron whose six edges have lengths 10, 20, 30, 40, 50 and 60 units...

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Qqq..cubed

Stage: 4 Challenge Level: Challenge Level:1

It is known that the area of the largest equilateral triangular section of a cube is 140sq cm. What is the side length of the cube? The distances between the centres of two adjacent faces of. . . .

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Investigating Solids with Face-transitivity

Stage: 4 and 5

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

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Bond Angles

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Think about the bond angles occurring in a simple tetrahedral molecule and ammonia.

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Interpenetrating Solids

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

This problem provides training in visualisation and representation of 3D shapes. You will need to imagine rotating cubes, squashing cubes and even superimposing cubes!

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Paper Folding - Models of the Platonic Solids

Stage: 2, 3 and 4

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

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Sliced

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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?

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Tetrahedra Tester

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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?

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Classifying Solids Using Angle Deficiency

Stage: 3 and 4 Challenge Level: Challenge Level:1

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

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Icosian Game

Stage: 3 Challenge Level: Challenge Level:1

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.

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Tetra Square

Stage: 3 Challenge Level: Challenge Level:1

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.

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A Mean Tetrahedron

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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?

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Tetrahedron Faces

Stage: 2 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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?

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Tetra Inequalities

Stage: 5 Challenge Level: Challenge Level:1

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.

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Reach for Polydron

Stage: 5 Challenge Level: Challenge Level:1

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.

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Four Points on a Cube

Stage: 5 Challenge Level: Challenge Level:1

What is the surface area of the tetrahedron with one vertex at O the vertex of a unit cube and the other vertices at the centres of the faces of the cube not containing O?