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. . . .
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.
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
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?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
Can you find a general rule for finding the areas of equilateral
triangles drawn on an isometric grid?
A circle is inscribed in a triangle which has side lengths of 8, 15
and 17 cm. What is the radius of the circle?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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?
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 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 is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?
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
Determine the total shaded area of the 'kissing triangles'.
Can you maximise the area available to a grazing goat?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
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.
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 work out the area of the inner square and give an
explanation of how you did it?
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. . . .
Three rods of different lengths form three sides of an enclosure
with right angles between them. What arrangement maximises the area
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
A point P is selected anywhere inside an equilateral triangle. What
can you say about the sum of the perpendicular distances from P to
the sides of the triangle? Can you prove your conjecture?
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. . . .
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
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 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
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.
This article, written for teachers, discusses the merits of
different kinds of resources: those which involve exploration and
those which centre on calculation.
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?
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. . . .
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. . . .
A task which depends on members of the group noticing the needs of
others and responding.
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
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
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.
Prove that the area of a quadrilateral is given by half the product of the lengths of the diagonals multiplied by the sine of the angle between the diagonals.
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
A follow-up activity to Tiles in the Garden.
Derive a formula for finding the area of any kite.
Explore one of these five pictures.
What is the same and what is different about these circle
questions? What connections can you make?
Which is a better fit, a square peg in a round hole or a round peg
in a square hole?
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. . . .
In the four examples below identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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