This is part of our collection of Short Problems.
You may also be interested in our longer problems on Pythagoras' Theorem and Trigonometry.
Printable worksheets containing selections of these problems are available here:
| Pythagoras Stage 4 ★ | Sheet 1 | Solutions | Pythagoras Stage 4 ★★★ | Sheet 1 | Solutions | |
| Pythagoras Stage 4 ★★ | Sheet 1 | Solutions | Trigonometry Stage 4 ★★★ | Sheet 1 | Solutions | |
| Sheet 2 | Solutions | |||||
problem
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?
problem
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?
problem
Pythagoras' Dream
Age
14 to 16
Challenge level
Can you work out the area of this isosceles right angled triangle?
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.
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?
problem
Rectangle Rearrangement
Age
14 to 16
Challenge level
A 3x8 rectangle is cut into two pieces... then rearranged to form a right-angled triangle. What is the perimeter of the triangle formed?
problem
Tetromino Diagonal
Age
14 to 16
Challenge level
Can you calculate the length of this diagonal line?
problem
Folding in Half
Age
14 to 16
Challenge level
How does the perimeter change when we fold this isosceles triangle in half?
problem
Unusual Polygon
Age
14 to 16
Challenge level
What is the perimeter of this unusually shaped polygon...
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?
problem
Winding vine
Age
14 to 16
Challenge level
A vine is growing up a pole. Can you find its length?
problem
Question of Three Sides
Age
14 to 16
Challenge level
Can you find the length of the third side of this triangle?
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?
problem
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?
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?
problem
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?
problem
Arc Radius
Age
14 to 16
Challenge level
Two arcs are drawn in a right-angled triangle as shown. What is the length $r$?
problem
Unusual Quadrilateral
Age
14 to 16
Challenge level
This quadrilateral has an unusual shape. Are you able to find its area?
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?
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?
problem
Diagonal area
Age
14 to 16
Challenge level
A square has area 72 cm$^2$. Find the length of its diagonal.
problem
Three right angles
Age
14 to 16
Challenge level
Work your way through these right-angled triangles to find $x$.
problem
Indigo Interior
Age
14 to 16
Challenge level
The diagram shows 8 shaded squares inside a circle. What is the shaded area?
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?
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?
problem
Triple Pythagoras
Age
14 to 16
Challenge level
Can you work out the length of the diagonal of the cuboid?
problem
Integers on a Sphere
Age
14 to 16
Challenge level
Can you find all the integer coordinates on a sphere of radius 3?
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?
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?
problem
Four circles
Age
14 to 16
Challenge level
Can you find the radius of the larger circle in the diagram?
problem
Smartphone Screen
Age
14 to 16
Challenge level
Can you find the length and width of the screen of this smartphone in inches?
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?
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?
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?
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?
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?
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.
problem
Height of the Tower
Age
14 to 16
Challenge level
How do these measurements enable you to find the height of this tower?
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?
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?
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?
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?
problem
Centre Square
Age
14 to 16
Challenge level
What does Pythagoras' Theorem tell you about the radius of these circles?