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Broad Topics > Measures and Mensuration > Area

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Growing Rectangles

Stage: 3 Challenge Level: Challenge Level:1

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

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Curvy Areas

Stage: 4 Challenge Level: Challenge Level:1

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

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Maths Filler

Stage: 4 Challenge Level: Challenge Level:1

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

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Can They Be Equal?

Stage: 3 Challenge Level: Challenge Level:1

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

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Partly Circles

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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

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All in a Jumble

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Gutter

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Wallpaper

Stage: 1 Challenge Level: Challenge Level:2 Challenge Level:2

These pieces of wallpaper need to be ordered from smallest to largest. Can you find a way to do it?

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Torn Shapes

Stage: 2 Challenge Level: Challenge Level:1

These rectangles have been torn. How many squares did each one have inside it before it was ripped?

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Sizing Them Up

Stage: 1 Challenge Level: Challenge Level:1

Can you put these shapes in order of size? Start with the smallest.

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Trapezium Four

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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More Transformations on a Pegboard

Stage: 2 Challenge Level: Challenge Level:2 Challenge Level:2

Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

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Spokes

Stage: 5 Challenge Level: Challenge Level:1

Draw three equal line segments in a unit circle to divide the circle into four parts of equal area.

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Areas of Parallelograms

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Warmsnug Double Glazing

Stage: 3 Challenge Level: Challenge Level:1

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

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Isosceles Triangles

Stage: 3 Challenge Level: Challenge Level:1

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?

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Fence It

Stage: 3 Challenge Level: Challenge Level:1

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

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Salinon

Stage: 4 Challenge Level: Challenge Level:1

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?

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Tilted Squares

Stage: 3 Challenge Level: Challenge Level:1

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

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Circle-in

Stage: 4 Challenge Level: Challenge Level:1

A circle is inscribed in a triangle which has side lengths of 8, 15 and 17 cm. What is the radius of the circle?

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An Unusual Shape

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Can you maximise the area available to a grazing goat?

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Inscribed in a Circle

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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?

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Pythagoras for a Tetrahedron

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

In a right-angled tetrahedron prove that the sum of the squares of the areas of the 3 faces in mutually perpendicular planes equals the square of the area of the sloping face. A generalisation. . . .

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Pick's Theorem

Stage: 3 Challenge Level: Challenge Level:1

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.

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Semi-detached

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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.

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Fitted

Stage: 2 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Nine squares with side lengths 1, 4, 7, 8, 9, 10, 14, 15, and 18 cm can be fitted together to form a rectangle. What are the dimensions of the rectangle?

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Transformations on a Pegboard

Stage: 2 Challenge Level: Challenge Level:1

How would you move the bands on the pegboard to alter these shapes?

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Numerically Equal

Stage: 2 Challenge Level: Challenge Level:2 Challenge Level:2

Can you draw a square in which the perimeter is numerically equal to the area?

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Of All the Areas

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Compare Areas

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Doesn't Add Up

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Quadarc

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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. . . .

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So Big

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

One side of a triangle is divided into segments of length a and b by the inscribed circle, with radius r. Prove that the area is: abr(a+b)/ab-r^2

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Areas and Ratios

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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.

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Mean Geometrically

Stage: 5 Challenge Level: Challenge Level:1

A and B are two points on a circle centre O. Tangents at A and B cut at C. CO cuts the circle at D. What is the relationship between areas of ADBO, ABO and ACBO?

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Making Boxes

Stage: 2 Challenge Level: Challenge Level:2 Challenge Level:2

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

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Shape Draw

Stage: 2 Challenge Level: Challenge Level:1

Use the information on these cards to draw the shape that is being described.

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Through the Window

Stage: 2 Challenge Level: Challenge Level:1

My local DIY shop calculates the price of its windows according to the area of glass and the length of frame used. Can you work out how they arrived at these prices?

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Dicey Perimeter, Dicey Area

Stage: 2 Challenge Level: Challenge Level:1

In this game for two players, you throw two dice and find the product. How many shapes can you draw on the grid which have that area or perimeter?

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Ribbon Squares

Stage: 2 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?

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Cover the Tray

Stage: 2 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

These practical challenges are all about making a 'tray' and covering it with paper.

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Perimeter Possibilities

Stage: 3 Challenge Level: Challenge Level:1

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

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Different Sizes

Stage: 1 and 2 Challenge Level: Challenge Level:1

A simple visual exploration into halving and doubling.

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Tiling Into Slanted Rectangles

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

A follow-up activity to Tiles in the Garden.

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Tiles in the Garden

Stage: 2 Challenge Level: Challenge Level:1

How many tiles do we need to tile these patios?

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Changing Areas, Changing Perimeters

Stage: 3 Challenge Level: Challenge Level:1

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?

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Cylinder Cutting

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

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

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Place Your Orders

Stage: 3 Challenge Level: Challenge Level:1

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

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Biology Measurement Challenge

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Shaping It

Stage: 1 and 2 Challenge Level: Challenge Level:1

These pictures were made by starting with a square, finding the half-way point on each side and joining those points up. You could investigate your own starting shape.