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Broad Topics > Measures and Mensuration > Area

Semi-detached

Stage: 4 Challenge Level:

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.

Rati-o

Stage: 3 Challenge Level:

Points P, Q, R and S each divide the sides AB, BC, CD and DA respectively in the ratio of 2 : 1. Join the points. What is the area of the parallelogram PQRS in relation to the original rectangle?

Hallway Borders

Stage: 3 Challenge Level:

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.

All in a Jumble

Stage: 3 Challenge Level:

My measurements have got all jumbled up! Swap them around and see if you can find a combination where every measurement is valid.

Pick's Theorem

Stage: 3 Challenge Level:

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.

Areas of Parallelograms

Stage: 4 Challenge Level:

Can you find the area of a parallelogram defined by two vectors?

Inscribed in a Circle

Stage: 4 Challenge Level:

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?

Lying and Cheating

Stage: 3 Challenge Level:

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!

Rhombus in Rectangle

Stage: 4 Challenge Level:

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.

Trapezium Four

Stage: 4 Challenge Level:

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

Isosceles

Stage: 3 Challenge Level:

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.

Max Box

Stage: 4 Challenge Level:

Three rods of different lengths form three sides of an enclosure with right angles between them. What arrangement maximises the area

Square Pizza

Stage: 4 Challenge Level:

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?

The Pi Are Square

Stage: 3 Challenge Level:

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?

Overlap

Stage: 3 Challenge Level:

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. . . .

Towers

Stage: 3 Challenge Level:

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?

Exploration Versus Calculation

Stage: 1, 2 and 3

This article, written for teachers, discusses the merits of different kinds of resources: those which involve exploration and those which centre on calculation.

Equilateral Areas

Stage: 4 Challenge Level:

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.

Six Discs

Stage: 4 Challenge Level:

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?

Salinon

Stage: 4 Challenge Level:

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?

An Unusual Shape

Stage: 3 Challenge Level:

Can you maximise the area available to a grazing goat?

Cylinder Cutting

Stage: 2 and 3 Challenge Level:

An activity for high-attaining learners which involves making a new cylinder from a cardboard tube.

Stage: 3 Challenge Level:

Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings?

Extending Great Squares

Stage: 2 and 3 Challenge Level:

Explore one of these five pictures.

Changing Areas, Changing Perimeters

Stage: 3 Challenge Level:

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?

Curvy Areas

Stage: 4 Challenge Level:

Have a go at creating these images based on circles. What do you notice about the areas of the different sections?

Biology Measurement Challenge

Stage: 4 Challenge Level:

Analyse these beautiful biological images and attempt to rank them in size order.

Tiling Into Slanted Rectangles

Stage: 2 and 3 Challenge Level:

A follow-up activity to Tiles in the Garden.

Towering Trapeziums

Stage: 4 Challenge Level:

Can you find the areas of the trapezia in this sequence?

Uniform Units

Stage: 4 Challenge Level:

Can you choose your units so that a cube has the same numerical value for it volume, surface area and total edge length?

Perimeter Possibilities

Stage: 3 Challenge Level:

I'm thinking of a rectangle with an area of 24. What could its perimeter be?

Growing Rectangles

Stage: 3 Challenge Level:

What happens to the area and volume of 2D and 3D shapes when you enlarge them?

Making Rectangles

Stage: 2 and 3 Challenge Level:

A task which depends on members of the group noticing the needs of others and responding.

Isosceles Triangles

Stage: 3 Challenge Level:

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

Fence It

Stage: 3 Challenge Level:

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

Tilted Squares

Stage: 3 Challenge Level:

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

Efficient Packing

Stage: 4 Challenge Level:

How efficiently can you pack together disks?

Partly Circles

Stage: 4 Challenge Level:

What is the same and what is different about these circle questions? What connections can you make?

Maths Filler

Stage: 3 Challenge Level:

Imagine different shaped vessels being filled. Can you work out what the graphs of the water level should look like?

Maths Filler 2

Stage: 4 Challenge Level:

Can you draw the height-time chart as this complicated vessel fills with water?

Stage: 4 Challenge Level:

Investigate the properties of quadrilaterals which can be drawn with a circle just touching each side and another circle just touching each vertex.

Great Squares

Stage: 2 and 3 Challenge Level:

Investigate how this pattern of squares continues. You could measure lengths, areas and angles.

Percentage Unchanged

Stage: 4 Challenge Level:

If the base of a rectangle is increased by 10% and the area is unchanged, by what percentage (exactly) is the width decreased by ?

Square Areas

Stage: 3 Challenge Level:

Can you work out the area of the inner square and give an explanation of how you did it?

Dividing the Field

Stage: 4 Challenge Level:

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. . . .

Stage: 4 Challenge Level:

In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?

Diagonals for Area

Stage: 4 Challenge Level:

Can you prove this formula for finding the area of a quadrilateral from its diagonals?

Two Circles

Stage: 4 Challenge Level:

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?

Dissect

Stage: 3 Challenge Level:

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?

From All Corners

Stage: 4 Challenge Level:

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.