Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
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. . . .
What fractions of the largest circle are the two shaded regions?
Explore one of these five pictures.
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
What happens to the area and volume of 2D and 3D shapes when you
A follow-up activity to Tiles in the Garden.
What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.
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 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
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?
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
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).
A task which depends on members of the group noticing the needs of
others and responding.
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
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.
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
How efficiently can you pack together disks?
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. . . .
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?
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!
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. . . .
Determine the total shaded area of the 'kissing triangles'.
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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
Derive a formula for finding the area of any kite.
Can you work out the area of the inner square and give an
explanation of how you did it?
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.
Given a square ABCD of sides 10 cm, and using the corners as
centres, construct four quadrants with radius 10 cm each inside the
square. The four arcs intersect at P, Q, R and S. Find the. . . .
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. . . .
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
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?
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?
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
Can you maximise the area available to a grazing goat?
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
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
What is the same and what is different about these circle
questions? What connections can you make?
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.
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
Imagine different shaped vessels being filled. Can you work out
what the graphs of the water level should look like?
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.
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
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.
Can you draw the height-time chart as this complicated vessel fills
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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.
Investigate the properties of quadrilaterals which can be drawn
with a circle just touching each side and another circle just
touching each vertex.