Perimeter, Area and Volume - Stage 3

Blue and White

KS 3 Challenge Level:

Challenge Level:1

Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?

Pick's Theorem

KS 3 Challenge Level:

Challenge Level:2 Challenge Level:2

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

An Unusual Shape

KS 3 Challenge Level:

Challenge Level:2 Challenge Level:2

Can you maximise the area available to a grazing goat?

Painted Cube

KS 3 Challenge Level:

Challenge Level:3 Challenge Level:3 Challenge Level:3

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?

Cuboids

KS 3 Challenge Level:

Challenge Level:3 Challenge Level:3 Challenge Level:3

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

On the Edge

KS 3 Challenge Level:

Challenge Level:2 Challenge Level:2

If you move the tiles around, can you make squares with different coloured edges?

Sending a Parcel

KS 3 Challenge Level:

Challenge Level:2 Challenge Level:2

What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?

Fence It

KS 3 Challenge Level:

Challenge Level:1

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

Efficient Cutting

KS 3 Challenge Level:

Challenge Level:3 Challenge Level:3 Challenge Level:3

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.

Isosceles Triangles

KS 3 Challenge Level:

Challenge Level:1

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

Warmsnug Double Glazing

KS 3 Challenge Level:

Challenge Level:3 Challenge Level:3 Challenge Level:3

How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?

Cola Can

KS 3 Challenge Level:

Challenge Level:3 Challenge Level:3 Challenge Level:3

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

Can They Be Equal?

KS 3 Challenge Level:

Challenge Level:1

Can you find rectangles where the value of the area is the same as the value of the perimeter?

Cuboid Challenge

KS 3 Challenge Level:

Challenge Level:2 Challenge Level:2

What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?

Changing Areas, Changing Perimeters

KS 3 Challenge Level:

Challenge Level:1

How can you change the area of a shape but keep its perimeter the same? How can you change the perimeter but keep the area the same?

Changing Areas, Changing Volumes

KS 3 Challenge Level:

Challenge Level:2 Challenge Level:2

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?

Perimeter Possibilities

KS 3 Challenge Level:

Challenge Level:1

I'm thinking of a rectangle with an area of 24. What could its perimeter be?

Triangles in a Square

KS 3 Challenge Level:

Challenge Level:1

What are the possible areas of triangles drawn in a square?

Perimeter Challenge

KS 3 Challenge Level:

Challenge Level:1

Can you deduce the perimeters of the shapes from the information given?

Colourful Cube

KS 3 Challenge Level:

Challenge Level:1

A colourful cube is made from little red and yellow cubes. But can you work out how many of each?

Cuboid Faces

KS 3 Short Challenge Level:

Challenge Level:1

Can you find the volume of a cuboid, given the areas of its faces?

Hawaiian Earring

KS 3 Short Challenge Level:

Challenge Level:2 Challenge Level:2

What fraction of the larger circle is outside the smaller circle?

Scalene Area

KS 3 Short Challenge Level:

Challenge Level:2 Challenge Level:2

Can you find the area of this scalene triangle?

Tilted Tank

KS 3 Short Challenge Level:

Challenge Level:2 Challenge Level:2

Can you find the height of the water in this tilted tank when it is flat?

Giant Rubik's Cube

KS 3 Short Challenge Level:

Challenge Level:1

How many cubes would be visible in a 12 by 12 by 12 Rubik’s cube?

Dividing a Square

KS 3 Short Challenge Level:

Challenge Level:2 Challenge Level:2

A square is divided into three shapes which all have equal areas. Can you find the length of this side?

Rectangle Split

KS 3 Short Challenge Level:

Challenge Level:3 Challenge Level:3 Challenge Level:3

Draw another line through the centre of this rectangle to split it into 4 pieces of equal area.

Semicircle Shape

KS 3 Short Challenge Level:

Challenge Level:2 Challenge Level:2

Find the shaded area of these shapes with perimeters made of semicircles.