Filter by: Content type: ALL Problems Articles Games Stage: All Stage 1&2 Stage 2&3 Stage 3&4 Stage 4&5 Challenge level:
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 a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
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.
Three rods of different lengths form three sides of an enclosure with right angles between them. What arrangement maximises the area
A circle is inscribed in a triangle which has side lengths of 8, 15 and 17 cm. What is the radius of the circle?
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. . . .
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?
Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?
Investigate the properties of quadrilaterals which can be drawn with a circle just touching each side and another circle just touching each vertex.
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 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. . . .
If the base of a rectangle is increased by 10% and the area is unchanged, by what percentage (exactly) is the width decreased by ?
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.
Four quadrants are drawn centred at the vertices of a square . Find the area of the central region bounded by the four arcs.
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?
In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?
If I print this page which shape will require the more yellow ink?
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?
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?
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?
Determine the total shaded area of the 'kissing triangles'.
Can you find the area of a parallelogram defined by two vectors?
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 out.
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.
The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?
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?
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?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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?
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.
This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.
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.
A follow-up activity to Tiles in the Garden.
Derive a formula for finding the area of any kite.
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.
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.
Can you work out the area of the inner square and give an explanation of how you did it?
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.
Can you maximise the area available to a grazing goat?
Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?
Which is a better fit, a square peg in a round hole or a round peg in a square hole?
An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.
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?
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
What fractions of the largest circle are the two shaded regions?
A task which depends on members of the group noticing the needs of others and responding.
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?