Three rods of different lengths form three sides of an enclosure with right angles between them. What arrangement maximises the area
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?
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 prove this formula for finding the area of a quadrilateral from its diagonals?
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. . . .
Can you find the areas of the trapezia in this sequence?
Four quadrants are drawn centred at the vertices of a square . Find the area of the central region bounded by the four arcs.
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.
Investigate the properties of quadrilaterals which can be drawn with a circle just touching each side and another circle just touching each vertex.
Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?
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?
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 find the area of a parallelogram defined by two vectors?
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?
If the base of a rectangle is increased by 10% and the area is unchanged, by what percentage (exactly) is the width decreased by ?
If I print this page which shape will require the more yellow ink?
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?
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.
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 into?
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?
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?
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.
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?
The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?
In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?
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. . . .
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.
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.
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).
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?
Can you draw the height-time chart as this complicated vessel fills with water?
Determine the total shaded area of the 'kissing triangles'.
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!
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. . . .
Can you maximise the area available to a grazing goat?
How efficiently can you pack together disks?
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?
Which is a better fit, a square peg in a round hole or a round peg in a square hole?
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?
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?
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?
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. . . .
Can you find rectangles where the value of the area is the same as the value of the perimeter?
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 area.