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.
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
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?
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.
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?
Can you find a general rule for finding the areas of equilateral
triangles drawn on an isometric grid?
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.
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. . . .
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?
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?
Can you find the area of a parallelogram defined by two vectors?
Three rods of different lengths form three sides of an enclosure
with right angles between them. What arrangement maximises the area
What fractions of the largest circle are the two shaded regions?
Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?
What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.
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?
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
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?
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?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
What is the same and what is different about these circle
questions? What connections can you make?
A task which depends on members of the group noticing the needs of
others and responding.
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!
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
Determine the total shaded area of the 'kissing triangles'.
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. . . .
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.
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 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
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. . . .
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
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.
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
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
Derive a formula for finding the area of any kite.
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. . . .
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?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
Can you maximise the area available to a grazing goat?
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. . . .
Can you prove this formula for finding the area of a quadrilateral from its diagonals?
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
A follow-up activity to Tiles in the Garden.
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. . . .
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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 find rectangles where the value of the area is the same as the value of the perimeter?