Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?
My measurements have got all jumbled up! Swap them around and see
if you can find a combination where every measurement is valid.
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
How efficiently can you pack together disks?
What happens to the area and volume of 2D and 3D shapes when you
Can you draw the height-time chart as this complicated vessel fills
What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.
Explore one of these five pictures.
A follow-up activity to Tiles in the Garden.
Analyse these beautiful biological images and attempt to rank them in size order.
Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?
Derive a formula for finding the area of any kite.
Imagine different shaped vessels being filled. Can you work out
what the graphs of the water level should look like?
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 same and what is different about these circle
questions? What connections can you make?
Investigate the properties of quadrilaterals which can be drawn
with a circle just touching each side and another circle just
touching each vertex.
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. . . .
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. . . .
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?
A task which depends on members of the group noticing the needs of
others and responding.
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
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.
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!
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.
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?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
Can you maximise the area available to a grazing goat?
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
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?
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?
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?
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.
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.
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
In the four examples below identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
Can you work out the area of the inner square and give an
explanation of how you did it?
A white cross is placed symmetrically in a red disc with the central square of side length sqrt 2 and the arms of the cross of length 1 unit. What is the area of the disc still showing?
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
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
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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. . . .
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. . . .
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
Triangle ABC is right angled at A and semi circles are drawn on all three sides producing two 'crescents'. Show that the sum of the areas of the two crescents equals the area of triangle ABC.