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### Number and algebra

### Geometry and measure

### Probability and statistics

### Working mathematically

### For younger learners

### Advanced mathematics

# A Quartet of Tetrahedra

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Tetra Inequalities

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Tetra Slice

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Tetra Perp

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Pythagoras for a Tetrahedron

## You may also like

### Patterns in Number Sequences

### Reasoning Geometrically

### Reasoning with Numbers

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This feature is all about tetrahedra, with four problems for you to solve.

You can find another problem involving a tetrahedron in **STEP Support Programme Foundation Assignment 5**.

Age 16 to 18

Challenge Level

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

Can you prove that a quadrilateral drawn inside a tetrahedron is a parallelogram?

Age 16 to 18

Challenge Level

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

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.

These resources are designed to get you thinking about number sequences and patterns.

These resources are designed to get you thinking about geometrical reasoning.

These resources are designed to get you thinking about reasoning with numbers.