Three rods of different lengths form three sides of an enclosure with right angles between them. What arrangement maximises the area
Can you find the areas of the trapezia in this sequence?
Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?
Can you prove this formula for finding the area of a quadrilateral from its diagonals?
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 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?
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. . . .
The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?
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.
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 choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?
Four quadrants are drawn centred at the vertices of a square . Find the area of the central region bounded by the four arcs.
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?
Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
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 the area of a parallelogram defined by two vectors?
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.
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 properties of quadrilaterals which can be drawn with a circle just touching each side and another circle just touching each vertex.
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?
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?
In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?
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!
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.
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?
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?
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?
How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?
How efficiently can you pack together disks?
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?
I'm thinking of a rectangle with an area of 24. What could its perimeter be?
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.
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?
What is the same and what is different about these circle questions? What connections can you make?
Determine the total shaded area of the 'kissing triangles'.
Can you draw the height-time chart as this complicated vessel fills with water?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
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.
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.
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?
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. . . .
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Which is a better fit, a square peg in a round hole or a round peg in a square hole?
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.
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. . . .
Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?