What fractions of the largest circle are the two shaded regions?
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. . . .
Analyse these beautiful biological images and attempt to rank them in size order.
Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
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?
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.
A follow-up activity to Tiles in the Garden.
Explore one of these five pictures.
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 is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
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. . . .
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. . . .
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?
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. . . .
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.
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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. . . .
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?
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.
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.
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 find the area of a parallelogram defined by two vectors?
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 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 find rectangles where the value of the area is the same as the value of the perimeter?
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 maximise the area available to a grazing goat?
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 happens to the area and volume of 2D and 3D shapes when you
Imagine different shaped vessels being filled. Can you work out
what the graphs of the water level should look like?
Can you work out the area of the inner square and give an
explanation of how you did it?
Can you draw the height-time chart as this complicated vessel fills
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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?
Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?
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. . . .
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
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Can you find a general rule for finding the areas of equilateral
triangles drawn on an isometric grid?
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. . . .
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
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).
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. . . .
Straight lines are drawn from each corner of a square to the mid
points of the opposite sides. Express the area of the octagon that
is formed at the centre as a fraction of the area of the square.
How efficiently can you pack together disks?