How efficiently can you pack together disks?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Explore one of these five pictures.
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.
What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.
Analyse these beautiful biological images and attempt to rank them in size order.
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 follow-up activity to Tiles in the Garden.
Can you draw the height-time chart as this complicated vessel fills with water?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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.
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?
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.
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. . . .
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. . . .
Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?
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. . . .
Investigate how this pattern of squares continues. You could measure lengths, areas and angles.
Can you maximise the area available to a grazing goat?
In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
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?
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. . . .
Which is a better fit, a square peg in a round hole or a round peg in a square hole?
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
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 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. . . .
A task which depends on members of the group noticing the needs of others and responding.
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
What happens to the area and volume of 2D and 3D shapes when you enlarge them?
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?
Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?
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!
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?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
Can you find the area of a parallelogram defined by two vectors?
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?
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?
What is the same and what is different about these circle questions? What connections can you make?
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.
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?
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 work out the area of the inner square and give an explanation of how you did it?
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. . . .
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).
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.
Derive a formula for finding the area of any kite.
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?
What fractions of the largest circle are the two shaded regions?