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.

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

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?

Three squares are drawn on the sides of a triangle ABC. Their areas are respectively 18 000, 20 000 and 26 000 square centimetres. If the outer vertices of the squares are joined, three more. . . .

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.

The area of a square inscribed in a circle with a unit radius is, satisfyingly, 2. What is the area of a regular hexagon inscribed in a circle with a unit radius?

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

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

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?

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

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

A circle with the radius of 2.2 centimetres is drawn touching the sides of a square. What area of the square is NOT covered by the circle?

Given a square ABCD of sides 10 cm, and using the corners as centres, construct four quadrants with radius 10 cm each inside the square. The four arcs intersect at P, Q, R and S. Find the. . . .

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?

Investigate the properties of quadrilaterals which can be drawn with a circle just touching each side and another circle just touching each vertex.

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

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!

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

What fractions of the largest circle are the two shaded regions?

What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.

A hallway floor is tiled and each tile is one foot square. Given that the number of tiles around the perimeter is EXACTLY half the total number of tiles, find the possible dimensions of the hallway.

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

What is the same and what is different about these circle questions? What connections can you make?

This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.

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. . . .

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?

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?

A follow-up activity to Tiles in the Garden.

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. . . .

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

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.

Have a go at creating these images based on circles. What do you notice about the areas of the different sections?

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

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).

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. . . .

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.

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

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?

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.

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?

This rectangle is cut into five pieces which fit exactly into a triangular outline and also into a square outline where the triangle, the rectangle and the square have equal areas.

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?

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

What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?