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 how this pattern of squares continues. You could measure lengths, areas and angles.
A follow-up activity to Tiles in the Garden.
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?
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 overlap?
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.
Investigate the properties of quadrilaterals which can be drawn with a circle just touching each side and another circle just touching each vertex.
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?
Explore one of these five pictures.
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.
Three squares are drawn on the sides of a triangle ABC. Their areas are respectively 18 000, 20 000 and 26 000 square centimetres. If the outer vertices of the squares are joined, three more. . . .
What is the shape and dimensions of a box that will contain six cups and have as small a surface area as possible.
Four quadrants are drawn centred at the vertices of a square . Find the area of the central region bounded by the four arcs.
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
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 squares?
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. . . .
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?
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!
A task which depends on members of the group noticing the needs of others and responding.
What happens to the area and volume of 2D and 3D shapes when you enlarge them?
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?
How efficiently can you pack together disks?
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
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 same radius.
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.
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
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.
If the base of a rectangle is increased by 10% and the area is unchanged, by what percentage (exactly) is the width decreased by ?
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).
Which is a better fit, a square peg in a round hole or a round peg in a square hole?
Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?
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. . . .
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
Can you prove this formula for finding the area of a quadrilateral from its diagonals?
You have a 12 by 9 foot carpet with an 8 by 1 foot hole exactly in the middle. Cut the carpet into two pieces to make a 10 by 10 foot square carpet.
Derive a formula for finding the area of any kite.
In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?
A farmer has a field which is the shape of a trapezium as illustrated below. To increase his profits he wishes to grow two different crops. To do this he would like to divide the field into two. . . .
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. . . .
A trapezium is divided into four triangles by its diagonals. Suppose the two triangles containing the parallel sides have areas a and b, what is the area of the trapezium?
Determine the total shaded area of the 'kissing triangles'.
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?
Can you work out the area of the inner square and give an explanation of how you did it?
Can you maximise the area available to a grazing goat?
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.
What is the same and what is different about these circle questions? What connections can you make?
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.
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?