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. . . .
Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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?
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 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.
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Can you find the area of a parallelogram defined by two vectors?
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.
Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
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?
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
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 maximise the area available to a grazing goat?
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.
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.
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. . . .
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?
Can you find the areas of the trapezia in this sequence?
Can you work out the area of the inner square and give an explanation of how you did it?
Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?
Determine the total shaded area of the 'kissing triangles'.
Which is a better fit, a square peg in a round hole or a round peg in a square hole?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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.
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.
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 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. . . .
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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?
Three rods of different lengths form three sides of an enclosure with right angles between them. What arrangement maximises the area
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?
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?
What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.
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. . . .
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. . . .
A task which depends on members of the group noticing the needs of others and responding.
What happens to the area and volume of 2D and 3D shapes when you enlarge them?
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?
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. . . .
What is the same and what is different about these circle questions? What connections can you make?
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.
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 follow-up activity to Tiles in the Garden.
Explore one of these five pictures.