Perimeter, Area and Volume - Stage 4

list

Perimeter, Area and Volume Short Problems

Age

11 to 16

A collection of short problems on area and volume.

problem

F'arc'tion

Age

14 to 16

Challenge level

At the corner of the cube circular arcs are drawn and the area enclosed shaded. What fraction of the surface area of the cube is shaded? Try working out the answer without recourse to pencil and paper.

problem

Arclets

Age

14 to 16

Challenge level

Each of the following shapes is made from arcs of a circle of radius r. What is the perimeter of a shape with 3, 4, 5 and n "nodes".

problem

Triangles and petals

Age

14 to 16

Challenge level

An equilateral triangle rotates around regular polygons and produces an outline like a flower. What are the perimeters of the different flowers?

problem

Circle in a Semicircle

Age

14 to 16

Challenge level

Imagine cutting out a circle which is just contained inside a semicircle. What fraction of the semi-circle will remain?

problem

Pencil Turning

Age

14 to 16

Challenge level

Rotating a pencil twice about two different points gives surprising results...

problem

Annulus Area

Age

14 to 16

Challenge level

Weekly Problem 38 - 2011
Given three concentric circles, shade in the annulus formed by the smaller two. What percentage of the larger circle is now shaded?

problem

Maximised Area

Age

14 to 16

Challenge level

Of these five figures, which shaded area is the greatest? The large circle in each figure has the same radius.

problem

Canny Fraction

Age

14 to 16

Challenge level

What fraction of the volume of this can is filled with lemonade?

problem

Salinon

Age

14 to 16

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?

problem

Sticky Tape

Age

14 to 16

Challenge level

Work out the radius of a roll of adhesive tape.

problem

Roll On

Age

14 to 16

Challenge level

Weekly Problem 5 - 2006
How many times does the inside disc have to roll around the inside of the ring to return to its initial position?

problem

Running Race

Age

14 to 16

Challenge level

Weekly Problem 13 - 2006
If three runners run at the same constant speed around the race tracks, in which order do they finish?

problem

Semicircle Stack

Age

14 to 16

Challenge level

What is the total area enclosed by the three semicicles?

problem

Eyelids

Age

14 to 16

Challenge level

Can you work out the shaded area surrounded by these arcs?

problem

Square Ratio

Age

14 to 16

Challenge level

A square is divided into four rectangles and a square. Can you work out the ratio of the side lengths of the rectangles?

problem

Rolling Inside

Age

14 to 16

Challenge level

Weekly Problem 11 - 2007
A circle of radius 1 rolls without slipping round the inside of a square of side length 4. Find an expression for the number of revolutions the circle makes.

problem

Gutter

Age

14 to 16

Challenge level

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

problem

Funnel

Age

14 to 16

Challenge level

A plastic funnel is used to pour liquids through narrow apertures. What shape funnel would use the least amount of plastic to manufacture for any specific volume ?

problem

Immersion

Age

14 to 16

Challenge level

Various solids are lowered into a beaker of water. How does the water level rise in each case?

problem

Curvy areas

Age

14 to 16

Challenge level

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

problem

Rectangle Cutting

Age

14 to 16

Challenge level

Tom and Jerry start with identical sheets of paper. Each one cuts his sheet in a different way. Can you find the perimeter of the original sheet?

problem

Crazy Shading

Age

14 to 16

Challenge level

Can you work out the fraction of the larger square that is covered by the shaded area?

problem

Efficient Cutting Poster

Age

14 to 16

Challenge level

Efficient Cutting Poster

problem

Circled Corners

Age

14 to 16

Challenge level

Three circles have been drawn at the vertices of this triangle. What is the area of the inner shaded area?

problem

Wood Pile Perimeter

Age

14 to 16

Challenge level

Weekly Problem 30 - 2011
Three touching circles have an interesting area between them...

problem

On The Edge Poster

Age

11 to 14

Challenge level

On the Edge Poster

problem

Track design

Age

14 to 16

Challenge level

Where should runners start the 200m race so that they have all run the same distance by the finish?

problem

Fill Me Up Too

Age

14 to 16

Challenge level

In Fill Me Up we invited you to sketch graphs as vessels are filled with water. Can you work out the equations of the graphs?

problem

Curvy Areas Poster

Age

14 to 16

Challenge level

Curvy Areas Poster

problem

Arc Area

Age

14 to 16

Challenge level

What is the area of the shape enclosed by the line and arcs?

problem

Tadpoles

Age

14 to 16

Challenge level

The diagram shows a shaded shape bounded by circular arcs. What is the difference in area betweeen this and the equilateral triangle shown?

problem

Penny Farthing

Age

14 to 16

Challenge level

Boris' bicycle has a smaller back wheel than front wheel. Can you work out how many revolutions the front wheel made if the back wheel did 120,000?

problem

In or Out?

Age

14 to 16

Challenge level

Weekly Problem 52 - 2014
Four arcs are drawn in a circle to create a shaded area. What fraction of the area of the circle is shaded?

problem

Four Leaf Clover

Age

14 to 16

Challenge level

The diagram shows four equal discs and a square. What is the perimeter of the figure?

problem

Cut-Up Square

Age

14 to 16

Challenge level

Weekly Problem 15 - 2015
In the diagram, two lines have been drawn in a square. What is the ratio of the areas marked?

problem

Emptied Cube

Age

14 to 16

Challenge level

Weekly Problem 26 - 2015
What are the volume and surface area of this 'Cubo Vazado' or 'Emptied Cube'?

problem

Square Flower

Age

14 to 16

Challenge level

The diagram shows 8 circles surrounding a region. What is the perimeter of the shaded region?

problem

Trisected Triangle

Age

14 to 16

Challenge level

Weekly Problem 34 - 2015
Four tiles are given. For which of them can three be placed together to form an equilateral triangle?

problem

Clown Hats

Age

14 to 16

Challenge level

Weekly Problem 51 - 2015
Charlie is making clown hats from a piece of cardboard. What is the maximum number he can make?

problem

Smile

Age

14 to 16

Challenge level

Four semicircles are drawn on a line to form a shape. What is the area of this shape?

problem

Semicircular Design

Age

14 to 16

Challenge level

Weekly Problem 9 - 2016
The diagram to the right shows a logo made from semi-circular arcs. What fraction of the logo is shaded?

problem

Four Parts

Age

14 to 16

Challenge level

The circle of radius 4cm is divided into four congruent parts by arcs of radius 2cm as shown. What is the length of the perimeter of one of the parts, in cm?