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?
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.
Can you find the areas of the trapezia in this sequence?
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.
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. . . .
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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.
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?
Which is a better fit, a square peg in a round hole or a round peg in a square hole?
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?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Three rods of different lengths form three sides of an enclosure with right angles between them. What arrangement maximises the area
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.
Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
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.
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?
Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?
Can you maximise the area available to a grazing goat?
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 same and what is different about these circle questions? What connections can you make?
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. . . .
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. . . .
Investigate how this pattern of squares continues. You could measure lengths, areas and angles.
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?
Four quadrants are drawn centred at the vertices of a square . Find the area of the central region bounded by the four arcs.
A task which depends on members of the group noticing the needs of others and responding.
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?
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!
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?
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.
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.
What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.
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 article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
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?
Can you prove this formula for finding the area of a quadrilateral from its diagonals?
A follow-up activity to Tiles in the Garden.
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 work out the area of the inner square and give an explanation of how you did it?
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?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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.
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. . . .
Determine the total shaded area of the 'kissing triangles'.
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. . . .
Derive a formula for finding the area of any kite.
What happens to the area and volume of 2D and 3D shapes when you enlarge them?