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

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.

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?

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?

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. . . .

Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?

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

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.

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?

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.

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

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

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

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

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. . . .

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

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?

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?

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.

Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.

Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. . . .

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. . . .

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 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?

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.

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

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

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?

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

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.

Three rods of different lengths form three sides of an enclosure with right angles between them. What arrangement maximises the area

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

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

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?

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

You can move the 4 pieces of the jigsaw and fit them into both outlines. Explain what has happened to the missing one unit of area.

Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube made from 27 unit cubes so that the surface area of the remaining solid is the same as the surface area of the original 3 by 3 by 3. . . .

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

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. . . .

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).

If I print this page which shape will require the more yellow ink?

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!

A follow-up activity to Tiles in the Garden.

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

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?