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Broad Topics > Measures and Mensuration > Area

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?

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?

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?

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. . . .

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.

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.

Square Pegs

Stage: 3 Challenge Level:

Which is a better fit, a square peg in a round hole or a round peg in a square hole?

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.

Crescents and Triangles

Stage: 4 Challenge Level:

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.

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.

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. . . .

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. . . .

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.

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?

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?

Partly Circles

Stage: 4 Challenge Level:

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

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.

Semi-square

Stage: 4 Challenge Level:

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?

Same Height

Stage: 4 Challenge Level:

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?

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.

The Pillar of Chios

Stage: 3 Challenge Level:

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. . . .

Perimeter Possibilities

Stage: 3 Challenge Level:

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

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?

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. . . .

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.

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?

Diagonals for Area

Stage: 4 Challenge Level:

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

Compare Areas

Stage: 4 Challenge Level:

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

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?

Tiling Into Slanted Rectangles

Stage: 2 and 3 Challenge Level:

A follow-up activity to Tiles in the Garden.

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.

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?

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?

Extending Great Squares

Stage: 2 and 3 Challenge Level:

Explore one of these five pictures.

Growing Rectangles

Stage: 3 Challenge Level:

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

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?

Making Rectangles

Stage: 2 and 3 Challenge Level:

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

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?

Cylinder Cutting

Stage: 2 and 3 Challenge Level:

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

An Unusual Shape

Stage: 3 Challenge Level:

Can you maximise the area available to a grazing goat?

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.

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.

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!

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?

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 ?

Pie Cuts

Stage: 3 Challenge Level:

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).

Carpet Cuts

Stage: 3 Challenge Level:

You have a 12 by 9 foot carpet with an 8 by 1 foot hole exactly in the middle. Cut the carpet into two pieces to make a 10 by 10 foot square carpet.

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?

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?