Surface and surface area

article
When the angles of a triangle don't add up to 180 degrees

This article outlines the underlying axioms of spherical geometry giving a simple proof that the sum of the angles of a triangle on the surface of a unit sphere is equal to pi plus the area of the triangle.
problem
Four points on a cube

Age

16 to 18

Challenge level

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?
problem
All tied up

Age

14 to 16

Challenge level

A ribbon runs around a box so that it makes a complete loop with two parallel pieces of ribbon on the top. How long will the ribbon be?
problem
Inside out

Age

14 to 16

Challenge level

There are 27 small cubes in a 3 x 3 x 3 cube, 54 faces being visible at any one time. Is it possible to reorganise these cubes so that by dipping the large cube into a pot of paint three times you can colour every face of all of the smaller cubes?
problem
Uniform units

Age

14 to 16

Challenge level

Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?
problem
Take ten

Age

11 to 14

Challenge level

Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube so that the surface area of the remaining solid is the same as the surface area of the original?
problem
Cubic conundrum

Age

7 to 16

Challenge level

Which of the following cubes can be made from these nets?
problem
Peeling the apple or the cone that lost its head

Age

14 to 16

Challenge level

How much peel does an apple have?
problem
Cubes

Age

7 to 11

Challenge level

How many faces can you see when you arrange these three cubes in different ways?
problem
The big cheese

Age

7 to 11

Challenge level

Investigate the area of 'slices' cut off this cube of cheese. What would happen if you had different-sized block of cheese to start with?