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# Pythagoras' Theorem and Trigonometry - Short Problems

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:

### Right-angled Midpoints

### Right Angled Possibilities

### Rectangle Rearrangement

### Tetromino Diagonal

### Pythagoras' Dream

### Out of the Window

### Crane Arm

### Folded Over

### Hexagon Perimeter

### Walk the Plank

### Folding in Half

### Arc Radius

### Unusual Polygon

### Unusual Quadrilateral

### Winding Vine

### Snapped Palm Tree

### Symmetric Angles

### Strike a Chord

### Question of Three Sides

### Triangular Teaser

### Diagonal Area

### Three Right Angles

### Centre Square

### Integers on a Sphere

### Oh So Circular

### Salt's Mill

### Smartphone Screen

### Ice Cream Tangent

### Distance to the Corner

### Overlapping Ribbons

### Square Overlap

### Common Tangent

### Folded Rectangle

### Indigo Interior

### The Roller and the Triangle

### Semicircle in a Semicircle

### When the Boat Comes In

### One or Two

### Four Circles

### Circle Time

### Triple Pythagoras

### Interior Squares

### Height of the Tower

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

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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 | |||||

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge 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?

Age 14 to 16

ShortChallenge 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?

Age 14 to 16

ShortChallenge Level

Can you calculate the length of this diagonal line?

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge 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?

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge 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?

Age 14 to 16

ShortChallenge 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?

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge 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?

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge 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?

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

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

Age 14 to 16

ShortChallenge Level

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