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There are **25** NRICH Mathematical resources connected to **Surface and surface area**, you may find related items under Measuring and calculating with units.

Problem
Primary curriculum
Secondary curriculum
### Changing Areas, Changing Volumes

How can you change the surface area of a cuboid but keep its volume the same? How can you change the volume but keep the surface area the same?

Age 11 to 14

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
### Brush Loads

How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Cuboids

Can you find a cuboid that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### 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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Painted Cube

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Cubes

How many faces can you see when you arrange these three cubes in different ways?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Tiling Into Slanted Rectangles

A follow-up activity to Tiles in the Garden.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Uniform Units

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

Age 14 to 16

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
### All Wrapped Up

What is the largest cuboid you can wrap in an A3 sheet of paper?

Age 7 to 11

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
### Inside Out

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### All Tied Up

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?

Age 14 to 16

Challenge Level

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

Problem
Primary curriculum
Secondary curriculum
### Cubic Conundrum

Which of the following cubes can be made from these nets?

Age 7 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Take Ten

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Plutarch's Boxes

According to Plutarch, the Greeks found all the rectangles with integer sides, whose areas are equal to their perimeters. Can you find them? What rectangular boxes, with integer sides, have their surface areas equal to their volumes?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### F'arc'tion

At the corner of the cube circular arcs are drawn and the area enclosed shaded. What fraction of the surface area of the cube is shaded? Try working out the answer without recourse to pencil and paper.

Age 14 to 16

ShortChallenge Level

Problem
Primary curriculum
Secondary curriculum
### Four Points on a Cube

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?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Wrapping Presents

Choose a box and work out the smallest rectangle of paper needed to wrap it so that it is completely covered.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### The Big Cheese

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?

Age 7 to 11

Challenge Level