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

Which is a better fit, a square peg in a round hole or a round peg in a square hole?

A square of area 40 square cms is inscribed in a semicircle. Find the area of the square that could be inscribed in a circle of the same radius.

ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.

Draw two circles, each of radius 1 unit, so that each circle goes through the centre of the other one. What is the area of the overlap?

Prove that a triangle with sides of length 5, 5 and 6 has the same area as a triangle with sides of length 5, 5 and 8. Find other pairs of non-congruent isosceles triangles which have equal areas.

Investigate the different ways of cutting a perfectly circular pie into equal pieces using exactly 3 cuts. The cuts have to be along chords of the circle (which might be diameters).

Bluey-green, white and transparent squares with a few odd bits of shapes around the perimeter. But, how many squares are there of each type in the complete circle? Study the picture and make. . . .

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

Follow the instructions and you can take a rectangle, cut it into 4 pieces, discard two small triangles, put together the remaining two pieces and end up with a rectangle the same size. Try it!

Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . .

Six circular discs are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box. Can you put the boxes in order according to the areas of their bases?

Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.

We started drawing some quadrilaterals - can you complete them?

Determine the total shaded area of the 'kissing triangles'.

What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?

A white cross is placed symmetrically in a red disc with the central square of side length sqrt 2 and the arms of the cross of length 1 unit. What is the area of the disc still showing?

A task which depends on members of the group noticing the needs of others and responding.

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?

You have a 12 by 9 foot carpet with an 8 by 1 foot hole exactly in the middle. Cut the carpet into two pieces to make a 10 by 10 foot square carpet.

Can you work out the area of the inner square and give an explanation of how you did it?

A tower of squares is built inside a right angled isosceles triangle. The largest square stands on the hypotenuse. What fraction of the area of the triangle is covered by the series of squares?

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

What are the possible dimensions of a rectangular hallway if the number of tiles around the perimeter is exactly half the total number of tiles?

Prove that the shaded area of the semicircle is equal to the area of the inner circle.

A farmer has a field which is the shape of a trapezium as illustrated below. To increase his profits he wishes to grow two different crops. To do this he would like to divide the field into two. . . .

Can you prove this formula for finding the area of a quadrilateral from its diagonals?

The points P, Q, R and S are the midpoints of the edges of a convex quadrilateral. What do you notice about the quadrilateral PQRS as the convex quadrilateral changes?

A trapezium is divided into four triangles by its diagonals. Suppose the two triangles containing the parallel sides have areas a and b, what is the area of the trapezium?

What happens to the area and volume of 2D and 3D shapes when you enlarge them?

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

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

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

Four rods, two of length a and two of length b, are linked to form a kite. The linkage is moveable so that the angles change. What is the maximum area of the kite?

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

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

Seven small rectangular pictures have one inch wide frames. The frames are removed and the pictures are fitted together like a jigsaw to make a rectangle of length 12 inches. Find the dimensions of. . . .

If the base of a rectangle is increased by 10% and the area is unchanged, by what percentage (exactly) is the width decreased by ?

The points P, Q, R and S are the midpoints of the edges of a non-convex quadrilateral.What do you notice about the quadrilateral PQRS and its area?

It is possible to dissect any square into smaller squares. What is the minimum number of squares a 13 by 13 square can be dissected into?

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

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

Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?