Perimeter, Area and Volume Short Problems: Age 14-16

This is part of our collection of Short Problems.

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

We have put together a selection of these short problems as printable worksheets:

Printable worksheets: Age 14-16 ⭐   Printable worksheet: Age 14-16 ⭐⭐   Printable worksheet: Age 14-16 ⭐⭐⭐   Printable worksheet solutions


  • Small pepper seedlings in turquoise pots.
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    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?

  • Small pepper seedlings in turquoise pots.
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    Semicircle Stack

    Age
    14 to 16
    Challenge level

    What is the total area enclosed by the three semicicles?

  • Small pepper seedlings in turquoise pots.
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    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?

  • Small pepper seedlings in turquoise pots.
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    Arc Area

    Age
    14 to 16
    Challenge level

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

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    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?

  • Small pepper seedlings in turquoise pots.
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    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?

  • Small pepper seedlings in turquoise pots.
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    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?

  • Small pepper seedlings in turquoise pots.
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    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?

  • Small pepper seedlings in turquoise pots.
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    Bobbly Perimeter

    Age
    14 to 16
    Challenge level

    Find the perimeter of this shape made of semicircles

  • Small pepper seedlings in turquoise pots.
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    Ratio of Areas

    Age
    14 to 16
    Challenge level

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

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    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.

  • Small pepper seedlings in turquoise pots.
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    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?

  • Small pepper seedlings in turquoise pots.
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    Pencil Turning

    Age
    14 to 16
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    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.

  • Small pepper seedlings in turquoise pots.
    problem

    Canny Fraction

    Age
    14 to 16
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Sticky Tape

    Age
    14 to 16
    Challenge level

    Work out the radius of a roll of adhesive tape.

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    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?

  • Small pepper seedlings in turquoise pots.
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    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.

  • Small pepper seedlings in turquoise pots.
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    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?

  • Small pepper seedlings in turquoise pots.
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    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?

  • Small pepper seedlings in turquoise pots.
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    Wood Pile Perimeter

    Age
    14 to 16
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    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?

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    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?

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    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'?

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    Square Flower

    Age
    14 to 16
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    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?

  • Small pepper seedlings in turquoise pots.
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    Semicircle Distance

    Age
    14 to 16
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Half an Annulus

    Age
    14 to 16
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Two Paths

    Age
    14 to 16
    Challenge level

    Which of these two paths made of semicircles is shorter?

  • Small pepper seedlings in turquoise pots.
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    Cones and Spheres

    Age
    14 to 16
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Loo Roll Emergency

    Age
    14 to 16
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Candy Floss

    Age
    14 to 16
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Similar Cylinders

    Age
    14 to 16
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Two Circles

    Age
    14 to 16
    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?

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    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?

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    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?

  • Small pepper seedlings in turquoise pots.
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    Eyelids

    Age
    14 to 16
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    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?

  • Small pepper seedlings in turquoise pots.
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    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?

  • Small pepper seedlings in turquoise pots.
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    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?

  • Small pepper seedlings in turquoise pots.
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    Sinking Feeling

    Age
    14 to 16
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Ratio Cut

    Age
    14 to 16
    Challenge 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?

  • Small pepper seedlings in turquoise pots.
    problem

    Snake Eyes

    Age
    14 to 16
    Challenge level

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

  • Small pepper seedlings in turquoise pots.
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    Tilted Aquarium

    Age
    14 to 16
    Challenge level

    Can you find the depth of water in this aquarium?

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