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 â˜… | Sheet 1 | Solutions | Pythagoras Stage 4 â˜…â˜…â˜… | Sheet 1 | Solutions | |

Pythagoras Stage 4 â˜…â˜… | Sheet 1 | Solutions | Trigonometry Stage 4 â˜…â˜…â˜… | Sheet 1 | Solutions | |

Sheet 2 | Solutions | |||||

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### Right Angled Possibilities

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

A 3x8 rectangle is cut into two pieces... then rearranged to form a right-angled triangle. What is the perimeter of the triangle formed?

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### Crane Arm

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

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### Right-Angled Midpoints

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

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### Out of the Window

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

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### Folded Over

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

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### Hexagon Perimeter

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

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### Walk the Plank

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

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### Unusual Quadrilateral

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

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### Symmetric Angles

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

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### Triangular Teaser

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?

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### Snapped Palm Tree

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

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### Smartphone Screen

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

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### Distance to the corner

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

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### Common Tangent

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

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### Folded rectangle

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

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### Semicircle in a Semicircle

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

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### When the boat comes in

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

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

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### Oh so Circular

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

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

problem

### square overlap

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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### Circle Time

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

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### Indigo Interior

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

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### One or Two

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

problem

### Salt's Mill

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

### overlapping ribbons

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

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