Can you find rectangles where the value of the area is the same as the value of the perimeter?
My measurements have got all jumbled up! Swap them around and see
if you can find a combination where every measurement is valid.
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?
Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?
Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
Derive a formula for finding the area of any kite.
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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.
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?
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. . . .
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?
Can you maximise the area available to a grazing goat?
Can you find the area of a parallelogram defined by two vectors?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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?
Imagine different shaped vessels being filled. Can you work out
what the graphs of the water level should look like?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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. . . .
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?
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
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. . . .
Investigate how this pattern of squares continues. You could
measure lengths, areas and angles.
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
Can you draw the height-time chart as this complicated vessel fills
Can you work out the area of the inner square and give an
explanation of how you did it?
Explore one of these five pictures.
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?
Analyse these beautiful biological images and attempt to rank them in size order.
A follow-up activity to Tiles in the Garden.
What happens to the area and volume of 2D and 3D shapes when you
Determine the total shaded area of the 'kissing triangles'.
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.
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?
How efficiently can you pack together disks?
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 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
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. . . .
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
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!
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 area of the quadrilateral APOQ? Working on the building
blocks will give you some insights that may help you to work 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?
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 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
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.