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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 another cube is 8cms. What is the side length of this cube? Another cube has an edge length of 12cm. At each vertex a tetrahedron with three mutually perpendicular edges of length 4cm is sliced away. What is the surface area and volume of the remaining solid?

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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.

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The Spider and the Fly

A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?

Peeling the Apple or the Cone That Lost Its Head

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

First the headless cone :

If this sector had a radius of 10 units what would the area be?

If the sector is made into a cone by closing up the gap, what will be the base radius for that cone?

If the cone now suffers a horizontal cut which removes the top and reduces the height of what remains to half the original cone height, what curved surface area does that lower shape now have?

Now for the Apple Peel :

Not a very healthy colour apple this one :

Can you see that an apple can be approximated to a stack of "headless cones" ?

Not very tall ones!

And you could use that curved surface area calculation you did earlier to get an approximation for the total apple peel.

Maybe use a spreadsheet to get improved accuracy with more frustums (the name for headless cones!) ?

Anyway, what's the amount of peel on an apple whose diameter is 8cm?