Pythagoras' Theorem and Trigonometry Short Problems: Age 11-16

This is part of our collection of Short Problems.

You may also be interested in our longer problems on Pythagoras' Theorem and Trigonometry: Age 11-16.

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

Pythagoras printable worksheet: Age 14-16 ⭐   Pythagoras printable worksheets: Age 14-16 ⭐⭐   Pythagoras printable worksheet: Age 14-16 ⭐⭐⭐

Trigonometry printable worksheet: Age 14-16 ⭐⭐⭐   Printable worksheet solutions


  • Small pepper seedlings in turquoise pots.
    problem

    Out of the Window

    Age
    14 to 16
    Challenge level

    Find out how many pieces of hardboard of differing sizes can fit through a rectangular window.

  • Small pepper seedlings in turquoise pots.
    problem

    Right Angled Possibilities

    Age
    14 to 16
    Challenge level

    If two of the sides of a right-angled triangle are 5cm and 6cm long, how many possibilities are there for the length of the third side?

  • Small pepper seedlings in turquoise pots.
    problem

    Rectangle Rearrangement

    Age
    14 to 16
    Challenge level

    A 3×8 rectangle is cut into two pieces... then rearranged to form a right-angled triangle. What is the perimeter of the triangle formed?

  • Small pepper seedlings in turquoise pots.
    problem

    Tetromino Diagonal

    Age
    14 to 16
    Challenge level

    Can you calculate the length of this diagonal line?

  • Small pepper seedlings in turquoise pots.
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    Crane Arm

    Age
    14 to 16
    Challenge level

    A parallelogram is formed by joining together four equilateral triangles. What is the length of the longest diagonal?

  • Small pepper seedlings in turquoise pots.
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    Right-Angled Midpoints

    Age
    14 to 16
    Challenge level

    If the midpoints of the sides of a right angled triangle are joined, what is the perimeter of this new triangle?

  • Small pepper seedlings in turquoise pots.
    problem

    Pythagoras' Dream

    Age
    14 to 16
    Challenge level

    Can you work out the area of this isosceles right angled triangle?

  • Small pepper seedlings in turquoise pots.
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    Folded Over

    Age
    14 to 16
    Challenge level

    A rectangular piece of paper is folded. Can you work out one of the lengths in the diagram?

  • Small pepper seedlings in turquoise pots.
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    Walk the Plank

    Age
    14 to 16
    Challenge level

    A rectangular plank fits neatly inside a square frame when placed diagonally. What is the length of the plank?

  • Small pepper seedlings in turquoise pots.
    problem

    Unusual Quadrilateral

    Age
    14 to 16
    Challenge level

    This quadrilateral has an unusual shape. Are you able to find its area?

  • Small pepper seedlings in turquoise pots.
    problem

    Symmetric Angles

    Age
    14 to 16
    Challenge level

    This diagram has symmetry of order four. Can you use different geometric properties to find a particular length?

  • Small pepper seedlings in turquoise pots.
    problem

    Folding in Half

    Age
    14 to 16
    Challenge level

    How does the perimeter change when we fold this isosceles triangle in half?

  • Small pepper seedlings in turquoise pots.
    problem

    Unusual Polygon

    Age
    14 to 16
    Challenge level

    What is the perimeter of this unusually shaped polygon...

  • Small pepper seedlings in turquoise pots.
    problem

    Triangular Teaser

    Age
    14 to 16
    Challenge level

    Triangle T has sides of lengths 6, 5 and 5. Triangle U has sides of lengths 8, 5 and 5. What is the ratio of their areas?

  • Small pepper seedlings in turquoise pots.
    problem

    Winding Vine

    Age
    14 to 16
    Challenge level

    A vine is growing up a pole. Can you find its length?

  • Small pepper seedlings in turquoise pots.
    problem

    Question of Three Sides

    Age
    14 to 16
    Challenge level

    Can you find the length of the third side of this triangle?

  • Small pepper seedlings in turquoise pots.
    problem

    Building Blocks

    Age
    14 to 16
    Challenge level

    Can you find the length of AB in this diagram?

  • Small pepper seedlings in turquoise pots.
    problem

    Snapped Palm Tree

    Age
    14 to 16
    Challenge level

    A palm tree has snapped in a storm. What is the height of the piece that is still standing?

  • Small pepper seedlings in turquoise pots.
    problem

    Hexagon Perimeter

    Age
    14 to 16
    Challenge level

    A circle of radius 1 is inscribed in a regular hexagon. What is the perimeter of the hexagon?

  • Small pepper seedlings in turquoise pots.
    problem

    Arc Radius

    Age
    14 to 16
    Challenge level

    Two arcs are drawn in a right-angled triangle as shown. What is the length $r$?

  • Small pepper seedlings in turquoise pots.
    problem

    Strike a Chord

    Age
    14 to 16
    Challenge level

    Can you work out the radius of a circle from some information about a chord?

  • Small pepper seedlings in turquoise pots.
    problem

    Diagonal Area

    Age
    14 to 16
    Challenge level

    A square has area 72 cm$^2$. Find the length of its diagonal.

  • Small pepper seedlings in turquoise pots.
    problem

    Three Right Angles

    Age
    14 to 16
    Challenge level

    Work your way through these right-angled triangles to find $x$.

  • Small pepper seedlings in turquoise pots.
    problem

    Common Tangent

    Age
    14 to 16
    Challenge level

    Two circles touch, what is the length of the line that is a tangent to both circles?

  • Small pepper seedlings in turquoise pots.
    problem

    Semicircle in a Semicircle

    Age
    14 to 16
    Challenge level

    The diagram shows two semicircular arcs... What is the diameter of the shaded region?

  • Small pepper seedlings in turquoise pots.
    problem

    Interior Squares

    Age
    14 to 16
    Challenge level

    Calculate the ratio of areas of these squares which are inscribed inside a semi-circle and a circle.

  • Small pepper seedlings in turquoise pots.
    problem

    Oh so Circular

    Age
    14 to 16
    Challenge level

    The diagram shows two circles and four equal semi-circular arcs. The area of the inner shaded circle is 1. What is the area of the outer circle?

  • Small pepper seedlings in turquoise pots.
    problem

    Ice Cream Tangent

    Age
    14 to 16
    Challenge level

    The diagram shows a semi-circle and an isosceles triangle which have equal areas. What is the value of tan x?

  • Small pepper seedlings in turquoise pots.
    problem

    Circle Time

    Age
    14 to 16
    Challenge level

    Three circles of different radii each touch the other two. What can you deduce about the arc length between these points?

  • Small pepper seedlings in turquoise pots.
    problem

    Centre Square

    Age
    14 to 16
    Challenge level

    What does Pythagoras' Theorem tell you about the radius of these circles?

  • Small pepper seedlings in turquoise pots.
    problem

    Indigo Interior

    Age
    14 to 16
    Challenge level

    The diagram shows 8 shaded squares inside a circle. What is the shaded area?

  • Small pepper seedlings in turquoise pots.
    problem

    One or Two

    Age
    14 to 16
    Challenge level

    The diagrams show squares placed inside semicircles. What is the ratio of the shaded areas?

  • Small pepper seedlings in turquoise pots.
    problem

    Salt's Mill

    Age
    14 to 16
    Challenge level

    A window frame in Salt's Mill consists of two equal semicircles and a circle inside a large semicircle. What is the radius of the circle?

  • Small pepper seedlings in turquoise pots.
    problem

    Triple Pythagoras

    Age
    14 to 16
    Challenge level

    Can you work out the length of the diagonal of the cuboid?

  • Small pepper seedlings in turquoise pots.
    problem

    Integers on a Sphere

    Age
    14 to 16
    Challenge level

    Can you find all the integer coordinates on a sphere of radius 3?

  • Small pepper seedlings in turquoise pots.
    problem

    Overlapping Ribbons

    Age
    14 to 16
    Challenge level

    Two ribbons are laid over each other so that they cross. Can you find the area of the overlap?

  • Small pepper seedlings in turquoise pots.
    problem

    The Roller and the Triangle

    Age
    14 to 16
    Challenge level

    How much of the inside of this triangular prism can Clare paint using a cylindrical roller?

  • Small pepper seedlings in turquoise pots.
    problem

    Four Circles

    Age
    14 to 16
    Challenge level

    Can you find the radius of the larger circle in the diagram?

  • Small pepper seedlings in turquoise pots.
    problem

    Smartphone Screen

    Age
    14 to 16
    Challenge level

    Can you find the length and width of the screen of this smartphone in inches?

  • Small pepper seedlings in turquoise pots.
    problem

    Distance to the Corner

    Age
    14 to 16
    Challenge level

    Can you find the distance from the well to the fourth corner, given the distance from the well to the first three corners?

  • Small pepper seedlings in turquoise pots.
    problem

    Folded Rectangle

    Age
    14 to 16
    Challenge level

    Can you find the perimeter of the pentagon formed when this rectangle of paper is folded?

  • Small pepper seedlings in turquoise pots.
    problem

    Triangle Radius

    Age
    14 to 16
    Challenge level

    Can you find the radii of the small circles?

  • Small pepper seedlings in turquoise pots.
    problem

    When the Boat Comes In

    Age
    14 to 16
    Challenge level

    When you pull a boat in using a rope, does the boat move more quickly, more slowly, or at the same speed as you?

  • Small pepper seedlings in turquoise pots.
    problem

    Height of the Tower

    Age
    14 to 16
    Challenge level

    How do these measurements enable you to find the height of this tower?

  • Small pepper seedlings in turquoise pots.
    problem

    Diamond Ring

    Age
    14 to 16
    Challenge level

    Find the radius of the stone in this ring.

  • Small pepper seedlings in turquoise pots.
    problem

    Square Overlap

    Age
    14 to 16
    Challenge level

    The top square has been rotated so that the squares meet at a 60$^\text{o}$ angle. What is the area of the overlap?

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