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Broad Topics > Measures and Mensuration > Area

### Growing Rectangles

##### Stage: 3 Challenge Level:

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

### 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?

### Maths Filler

##### Stage: 4 Challenge Level:

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

### Can They Be Equal?

##### Stage: 3 Challenge Level:

Can you find rectangles where the value of the area is the same as the value of the perimeter?

### Partly Circles

##### Stage: 4 Challenge Level:

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

### 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.

### Gutter

##### Stage: 4 Challenge Level:

Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?

### 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?

### Areas of Parallelograms

##### Stage: 4 Challenge Level:

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

### Warmsnug Double Glazing

##### Stage: 3 Challenge Level:

How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?

### 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?

### 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?

### 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?

### An Unusual Shape

##### Stage: 3 Challenge Level:

Can you maximise the area available to a grazing goat?

### Inscribed in a Circle

##### Stage: 3 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?

### 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.

### 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.

### Of All the Areas

##### Stage: 4 Challenge Level:

Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?

### Compare Areas

##### Stage: 4 Challenge Level:

Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?

##### Stage: 4 Challenge Level:

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

##### Stage: 4 Challenge Level:

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. . . .

### Areas and Ratios

##### Stage: 4 Challenge Level:

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.

### Pebbles

##### Stage: 2 and 3 Challenge Level:

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

### Perimeter Possibilities

##### Stage: 3 Challenge Level:

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

### Tiling Into Slanted Rectangles

##### Stage: 2 and 3 Challenge Level:

A follow-up activity to Tiles in the Garden.

### 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?

### 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?

### Biology Measurement Challenge

##### Stage: 4 Challenge Level:

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

### Extending Great Squares

##### Stage: 2 and 3 Challenge Level:

Explore one of these five pictures.

### Making Rectangles

##### Stage: 2 and 3 Challenge Level:

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

##### 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.

### 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?

### Maths Filler 2

##### Stage: 4 Challenge Level:

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

### Efficient Packing

##### Stage: 4 Challenge Level:

How efficiently can you pack together disks?

### Two Shapes & Printer Ink

##### Stage: 4 Challenge Level:

If I print this page which shape will require the more yellow ink?

### 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.

### Shear Magic

##### Stage: 3 Challenge Level:

What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?

### 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?

### 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

### Squ-areas

##### Stage: 4 Challenge Level:

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. . . .

### Bound to Be

##### Stage: 4 Challenge Level:

Four quadrants are drawn centred at the vertices of a square . Find the area of the central region bounded by the four arcs.

### 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?

### Kite

##### Stage: 3 Challenge Level:

Derive a formula for finding the area of any kite.

### Appearing Square

##### Stage: 3 Challenge Level:

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. . . .

### Disappearing Square

##### Stage: 3 Challenge Level:

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. . . .

### Squaring the Circle

##### Stage: 3 Challenge Level:

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. . . .

### Take a Square

##### Stage: 4 Challenge Level:

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.

### 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. . . .