Area - squares and rectangles

problem
Favourite

Compare Areas

Age

14 to 16

Challenge level

Which has the greatest area, a circle or a square, inscribed in an isosceles right angle triangle?
problem
Favourite

Pythagoras for a Tetrahedron

Age

16 to 18

Challenge level

In a right-angled tetrahedron prove that the sum of the squares of the areas of the 3 faces in mutually perpendicular planes equals the square of the area of the sloping face. A generalisation of Pythagoras' Theorem.
problem

Take a Square

Age

14 to 16

Challenge level

Cut off three right angled isosceles triangles to produce a pentagon. With two lines, cut the pentagon into three parts which can be rearranged into another square.
problem

Paper Halving

Age

5 to 11

Challenge level

In how many ways can you halve a piece of A4 paper? How do you know they are halves?
problem

Lawn Border

Age

5 to 11

Challenge level

If I use 12 green tiles to represent my lawn, how many different ways could I arrange them? How many border tiles would I need each time?
problem

Cutting It Out

Age

5 to 11

Challenge level

I cut this square into two different shapes. What can you say about the relationship between them?
problem

Fit These Shapes

Age

5 to 11

Challenge level

What is the largest number of circles we can fit into the frame without them overlapping? How do you know? What will happen if you try the other shapes?
problem

Fencing Lambs

Age

7 to 11

Challenge level

A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.
problem

Great Squares

Age

7 to 11

Challenge level

Investigate how this pattern of squares continues. You could measure lengths, areas and angles.
problem

Two Squared

Age

7 to 11

Challenge level

What happens to the area of a square if you double the length of the sides? Try the same thing with rectangles, diamonds and other shapes. How do the four smaller ones fit into the larger one?