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### Number and algebra

### Geometry and measure

### Probability and statistics

### Working mathematically

### For younger learners

### Advanced mathematics

# Perimeter, Area and Volume Short Problems

### Chequered Cuboid

### Star in a Hexagon

### 3x3 Areas

### Mid-point Area

### Six Circles

### Hexa-split

### Giant Rubik's Cube

### Squares in a Square

### Cuboid Faces

### Cubic Masterpiece

### Double Cover

### Semicircle Shape

### Guillotine

### Tangram Area

### Tile Border

### Shaded End

### Open the Box

### Triangles' Triangle

### Strawberries and Peas

### Tilted Tank

### Cubes on a Cube

### Packing Small Boxes

### Hawaiian Earring

### Corner Cut

### Exactly Three-quarters

### L-emental

### Leaning Over

### Line of Squares

### Intersecting Squares

### Christmas Cut-out

### Margins

### Sideways Ratio

### Dividing a Square

### Triangle in a Hexagon

### Rectangle Split

### Pile Driver

### Four Square

### Hexagon Slices

### Four Parts

### Annulus Area

### Arc Area

### Semicircle Stack

### Bobbly Perimeter

### Ratio of Areas

### Penny Farthing

### Smile

### Crazy Shading

### Semicircular Design

### Pencil Turning

### Wood Pile Perimeter

### Semicircle Distance

### Trisected Triangle

### Rolling Inside

### Square in a Circle in a Square

### Maximised Area

### Four Leaf Clover

### Tadpoles

### Canny Fraction

### Loo Roll Emergency

### Rectangle Cutting

### Cones and Spheres

### F'arc'tion

### Half an Annulus

### Candy Floss

### Emptied Cube

### Circle in a Semicircle

### Two Paths

### Square Ratio

### Similar Cylinders

### Circled Corners

### Square Flower

### Ratio Cut

### In or Out?

### Running Race

### Snake Eyes

### Cut-up Square

### Sinking Feeling

### Clown Hats

### Eyelids

### Roll On

## You may also like

### Seven Squares - Group-worthy Task

Links to the University of Cambridge website
Links to the NRICH website Home page

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30 April (Primary), 1 May (Secondary)

30 April (Primary), 1 May (Secondary)

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This is part of our collection of Short Problems.

You may also be interested in our longer problems on Perimeter, Area and Volume Age 11-14 and Age 14-16.

Printable worksheets containing selections of these problems are available here:

Stage 3 ★ | Sheet 1 | Solutions | Stage 3 ★★ | Sheet 1 | Solutions | |

Sheet 2 | Solutions | Sheet 2 | Solutions | |||

Sheet 3 | Solutions | Sheet 3 | Solutions | |||

Sheet 4 | Solutions | |||||

Stage 3 ★★★ | Sheet 1 | Solutions | ||||

Stage 4 ★ | Sheet 1 | Solutions | Stage 4 ★★ | Sheet 1 | Solutions | |

Sheet 2 | Solutions | |||||

Sheet 3 | Solutions | Stage 4 ★★★ | Sheet 1 | Solutions |

Age 11 to 14

ShortChallenge Level

A large cuboid is made from cubes of equal size. What fraction of the surface area of the large cuboid is black?

Age 11 to 14

ShortChallenge Level

Weekly Problem 20 - 2017

The diagram shows a design formed by drawing six lines in a regular hexagon. What fraction of the hexagon is shaded?

Age 11 to 14

ShortChallenge Level

Which of the following shaded regions has an area different from the other shaded regions?

Age 11 to 14

ShortChallenge Level

M is the midpoint of the side of the rectangle. What is the area (in square units) of the triangle PMR?

Age 11 to 14

ShortChallenge Level

In the diagram, six circles of equal size touch adjacent circles and the sides of the large rectangle. What is the perimeter of the large rectangle?

Age 11 to 14

ShortChallenge Level

Which of the areas shown in the hexagons are equal to each other?

Age 11 to 14

ShortChallenge Level

How many cubes would be visible in a 12 by 12 by 12 Rubikâ€™s cube?

Age 11 to 14

ShortChallenge Level

In the diagram, the small squares are all the same size. What fraction of the large square is shaded?

Age 11 to 14

ShortChallenge Level

Can you find the volume of a cuboid, given the areas of its faces?

Age 11 to 14

ShortChallenge Level

Weekly Problem 49 - 2014

A blue cube is cut into 27 smaller cubes of equal size. What fraction of the total surface area of these cubes is blue?

Age 11 to 14

ShortChallenge Level

Weekly Problem 3 - 2007

What is the ratio of the area of the table covered twice, to the uncovered area?

Age 11 to 14

ShortChallenge Level

Find the shaded area of these shapes with perimeters made of semicircles.

Age 11 to 14

ShortChallenge Level

Weekly Problem 9 - 2006

What fraction of the area of the rectangle is shaded?

Age 11 to 14

ShortChallenge Level

Weekly Problem 29 - 2008

The seven pieces in this 12 cm by 12 cm square make a Tangram set. What is the area of the shaded parallelogram?

Age 11 to 14

ShortChallenge Level

What is the largest possible number of yellow tiles in this pattern?

Age 11 to 14

ShortChallenge Level

If the area of a face of a cuboid is one quarter of the area of each of the other two visible faces, what is the area of these faces?

Age 11 to 14

ShortChallenge Level

Weekly Problem 37 - 2015

A piece of card is folded to make an open box. Given its surface area, can you work out its volume?

Age 11 to 14

ShortChallenge Level

Weekly Problem 27 - 2009

The perimeter of a large triangle is 24 cm. What is the total length of the black lines used to draw the figure?

Age 11 to 14

ShortChallenge Level

Weekly Problem 17 - 2017

Yasmin lengthened one side of her pea bed by 3m to make it a square. This reduced her strawberry patch by $15m^2$. What was the original area of her pea bed?

Age 11 to 14

ShortChallenge Level

Can you find the height of the water in this tilted tank when it is flat?

Age 11 to 14

ShortChallenge Level

Weekly Problem 26 - 2016

A cube has each of its faces covered by one face of an identical cube, making a solid as shown. What is the surface area of the solid?

Age 11 to 14

ShortChallenge Level

How many small boxes will fit inside the big box?

Age 11 to 14

ShortChallenge Level

What fraction of the larger circle is outside the smaller circle?

Age 11 to 14

ShortChallenge Level

Weekly Problem 23 - 2017

Three small equilateral triangles of the same size are cut from the corners of a larger equilateral triangle. What is the side length of the small triangles?

Age 11 to 14

ShortChallenge Level

Weekly Problem 30 - 2007

Three-quarters of the area of the rectangle has been shaded. What is the length of x?

Age 11 to 14

ShortChallenge Level

Weekly Problem 28 - 2006

What can you say about the rectangles that form this L-shape?

Age 11 to 14

ShortChallenge Level

Weekly Problem 31 - 2017

The triangle HIJ has the same area as the square FGHI. What is the distance from J to the line extended through F and G?

Age 11 to 14

ShortChallenge Level

Weekly Problem 35 - 2016

What is the total perimeter of the squares, if the line GH in the diagram is 24cm?

Age 11 to 14

ShortChallenge Level

Weekly Problem 32 - 2014

Three overlapping squares are shown. If you know the areas of the overlapping and non-overlapping parts, can you work out the side lengths of the squares?

Age 11 to 14

ShortChallenge Level

Weekly Problem 52 - 2017

Sue cuts some squares from a piece of paper to make a Christmas decoration. What is the perimeter of the resulting shape?

Age 11 to 14

ShortChallenge Level

Weekly Problem 16 - 2008

A 30cm x 40cm page of a book includes a 2cm margin on each side... What percentage of the page is occupied by the margins?

Age 11 to 14

ShortChallenge Level

Weekly Problem 33 - 2014

A rectangle with area $125\text{cm}^2$ has sides in the ratio $4:5$. What is the perimeter of the rectangle?

Age 11 to 14

ShortChallenge Level

A square is divided into three shapes which all have equal areas. Can you find the length of this side?

Age 11 to 14

ShortChallenge Level

Weekly Problem 3 - 2009

What fraction of the area of this regular hexagon is the shaded triangle?

Age 11 to 14

ShortChallenge Level

Draw another line through the centre of this rectangle to split it into 4 pieces of equal area.

Age 11 to 14

ShortChallenge Level

Weekly Problem 38 - 2015

Where does the line through P that halves the figure shown meet the edge XY?

Age 11 to 14

ShortChallenge Level

Weekly Problem 17 - 2015

A square contains two overlapping squares. What is the total of the shaded regions?

Age 11 to 14

ShortChallenge Level

Weekly Problem 37 - 2007

This regular hexagon has been divided into four trapezia and one hexagon.... what is the ratio of the lengths of sides p, q and r?

Age 14 to 16

ShortChallenge 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?

Age 14 to 16

ShortChallenge 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?

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

What is the total area enclosed by the three semicicles?

Age 14 to 16

ShortChallenge Level

Find the perimeter of this shape made of semicircles

Age 14 to 16

ShortChallenge Level

What is the ratio of the area of the hexagon to the area of the triangle?

Age 14 to 16

ShortChallenge 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?

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge 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?

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

Weekly Problem 30 - 2011

Three touching circles have an interesting area between them...

Age 14 to 16

ShortChallenge Level

Can you find the shortest distance between the semicircles given the area between them?

Age 14 to 16

ShortChallenge Level

Weekly Problem 34 - 2015

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

Age 14 to 16

ShortChallenge 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.

Age 14 to 16

ShortChallenge Level

What is the ratio of the areas of the squares in the diagram?

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

When the roll of toilet paper is half as wide, what percentage of the paper is left?

Age 14 to 16

ShortChallenge 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?

Age 14 to 16

ShortChallenge Level

A solid metal cone is melted down and turned into spheres. How many spheres can be made?

Age 14 to 16

ShortChallenge 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.

Age 14 to 16

ShortChallenge Level

Can you locate the point on an annulus that splits it into two areas?

Age 14 to 16

ShortChallenge Level

What length of candy floss can Rita spin from her cylinder of sugar?

Age 14 to 16

ShortChallenge Level

Weekly Problem 26 - 2015

What are the volume and surface area of this 'Cubo Vazado' or 'Emptied Cube'?

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

Which of these two paths made of semicircles is shorter?

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

Two similar cylinders are formed from a block of metal. What is the volume of the smaller cylinder?

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

Cutting a rectangle from a corner to a point on the opposite side splits its area in the ratio 1:2. What is the ratio of a:b?

Age 14 to 16

ShortChallenge 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?

Age 14 to 16

ShortChallenge Level

Weekly Problem 13 - 2006

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

Age 14 to 16

ShortChallenge Level

Can you find the area of the yellow part of this snake's eye?

Age 14 to 16

ShortChallenge Level

Weekly Problem 15 - 2015

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

Age 14 to 16

ShortChallenge Level

Two vases are cylindrical in shape. Can you work out the original depth of the water in the larger vase?

Age 14 to 16

ShortChallenge Level

Weekly Problem 51 - 2015

Charlie is making clown hats from a piece of cardboard. What is the maximum number he can make?

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge 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?

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?