Can you find rectangles where the value of the area is the same as the value of the perimeter?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
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?
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?
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?
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. . . .
Can you maximise the area available to a grazing goat?
Can you find the area of a parallelogram defined by two vectors?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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.
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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 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 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?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
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?
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. . . .
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?
Four quadrants are drawn centred at the vertices of a square . Find
the area of the central region bounded by the four arcs.
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).
If I print this page which shape will require the more yellow ink?
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. . . .
Can you work out the area of the inner square and give an
explanation of how you did it?
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?
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?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
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 follow-up activity to Tiles in the Garden.
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?
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.
Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?
Explore one of these five pictures.
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?
What is the same and what is different about these circle
questions? What connections can you make?
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!
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
Investigate the properties of quadrilaterals which can be drawn
with a circle just touching each side and another circle just
touching each vertex.
A task which depends on members of the group noticing the needs of
others and responding.
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 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
What happens to the area and volume of 2D and 3D shapes when you
If the base of a rectangle is increased by 10% and the area is
unchanged, by what percentage (exactly) is the width decreased by ?
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?