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

My measurements have got all jumbled up! Swap them around and see if you can find a combination where every measurement is valid.

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.

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

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

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

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

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. . . .

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

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.

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?

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

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

Can you find the area of a parallelogram defined by two vectors?

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

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

Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?

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?

Make an eight by eight square, the layout is the same as a chessboard. You can print out and use the square below. What is the area of the square? Divide the square in the way shown by the red dashed. . . .

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

Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?

A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .

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

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

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

Can you draw the height-time chart as this complicated vessel fills with water?

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?

What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

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

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

Analyse these beautiful biological images and attempt to rank them in size order.

A follow-up activity to Tiles in the Garden.

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

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.

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.

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.

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. . . .

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

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!

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?

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?

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?

Points P, Q, R and S each divide the sides AB, BC, CD and DA respectively in the ratio of 2 : 1. Join the points. What is the area of the parallelogram PQRS in relation to the original rectangle?

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?

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.