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. . . .
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
Can you maximise the area available to a grazing goat?
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
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. . . .
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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.
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?
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. . . .
Can you work out the area of the inner square and give an
explanation of how you did it?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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. . . .
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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.
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. . . .
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?
Explore one of these five pictures.
How efficiently can you pack together disks?
What is the same and what is different about these circle
questions? What connections can you make?
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?
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.
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!
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?
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.
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 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. . . .
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 follow-up activity to Tiles in the Garden.
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
Determine the total shaded area of the 'kissing triangles'.
In this problem we are faced with an apparently easy area problem,
but it has gone horribly wrong! What happened?
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
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?
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
Can you find the area of a parallelogram defined by two vectors?
Can you prove this formula for finding the area of a quadrilateral from its diagonals?
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. . . .
Investigate the properties of quadrilaterals which can be drawn
with a circle just touching each side and another circle just
touching each vertex.
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 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?
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.
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
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
Derive a formula for finding the area of any kite.