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There are **32** NRICH Mathematical resources connected to **Spheres, cylinders and cones**, you may find related items under 3D geometry, shape and space.

Problem
Primary curriculum
Secondary curriculum
### Fill Me up Too

In Fill Me Up we invited you to sketch graphs as vessels are filled with water. Can you work out the equations of the graphs?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Which Solid?

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.

Age 7 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Immersion

Various solids are lowered into a beaker of water. How does the water level rise in each case?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Funnel

A plastic funnel is used to pour liquids through narrow apertures. What shape funnel would use the least amount of plastic to manufacture for any specific volume ?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Cola Can

An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Efficient Cutting

Use a single sheet of A4 paper and make a cylinder having the greatest possible volume. The cylinder must be closed off by a circle at each end.

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Sponge Sections

You have been given three shapes made out of sponge: a sphere, a cylinder and a cone. Your challenge is to find out how to cut them to make different shapes for printing.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Gym Bag

Can Jo make a gym bag for her trainers from the piece of fabric she has?

Age 11 to 16

Challenge Level

Project
Primary curriculum
Secondary curriculum
### Make Your Own Pencil Case

What shape would fit your pens and pencils best? How can you make it?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Cylinder Cutting

An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.

Age 7 to 11

Challenge Level

Article
Primary curriculum
Secondary curriculum
### Paint Rollers for Frieze Patterns.

Proofs that there are only seven frieze patterns involve complicated group theory. The symmetries of a cylinder provide an easier approach.

Age 11 to 16

Problem
Primary curriculum
Secondary curriculum
### Pack Man

A look at different crystal lattice structures, and how they relate to structural properties

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Packing 3D Shapes

What 3D shapes occur in nature. How efficiently can you pack these shapes together?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### There and Back Again

Bilbo goes on an adventure, before arriving back home. Using the information given about his journey, can you work out where Bilbo lives?

Age 11 to 14

Challenge Level

Article
Primary curriculum
Secondary curriculum
### Curvature of Surfaces

How do we measure curvature? Find out about curvature on soccer and rugby balls and on surfaces of negative curvature like banana skins.

Age 16 to 18

Problem
Primary curriculum
Secondary curriculum
### Spherical Triangles on Very Big Spheres

Shows that Pythagoras for Spherical Triangles reduces to Pythagoras's Theorem in the plane when the triangles are small relative to the radius of the sphere.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Pythagoras on a Sphere

Prove Pythagoras' Theorem for right-angled spherical triangles.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Flight Path

Use simple trigonometry to calculate the distance along the flight path from London to Sydney.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Peeling the Apple or the Cone That Lost Its Head

How much peel does an apple have?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Tin Tight

What's the most efficient proportion for a 1 litre tin of paint?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Air Routes

Find the distance of the shortest air route at an altitude of 6000 metres between London and Cape Town given the latitudes and longitudes. A simple application of scalar products of vectors.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Mesh

A spherical balloon lies inside a wire frame. How much do you need to deflate it to remove it from the frame if it remains a sphere?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Conical Bottle

A right circular cone is filled with liquid to a depth of half its vertical height. The cone is inverted. How high up the vertical height of the cone will the liquid rise?

Age 14 to 16

Challenge Level

Article
Primary curriculum
Secondary curriculum
### Conic Sections

The interplay between the two and three dimensional Euclidean geometry of conic sections is explored in this article. Suitable for students from 16+, teachers and parents.

Age 16 to 18

Article
Primary curriculum
Secondary curriculum
### 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.

Age 14 to 18

Article
Primary curriculum
Secondary curriculum
### Mouhefanggai

Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.

Age 14 to 16

Article
Primary curriculum
Secondary curriculum
### Volume of a Pyramid and a Cone

These formulae are often quoted, but rarely proved. In this article, we derive the formulae for the volumes of a square-based pyramid and a cone, using relatively simple mathematical concepts.

Age 11 to 14

Article
Primary curriculum
Secondary curriculum
### The Dodecahedron Explained

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

Age 16 to 18

Problem
Primary curriculum
Secondary curriculum
### In a Spin

What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Three Balls

A circle has centre O and angle POR = angle QOR. Construct tangents at P and Q meeting at T. Draw a circle with diameter OT. Do P and Q lie inside, or on, or outside this circle?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Ball Packing

If a ball is rolled into the corner of a room how far is its centre from the corner?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### 2D-3D

Two circles of equal size intersect and the centre of each circle is on the circumference of the other. What is the area of the intersection? Now imagine that the diagram represents two spheres of equal volume with the centre of each sphere on the surface of the other. What is the volume of intersection?

Age 16 to 18

Challenge Level