# 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:

 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

### Pythagoras' Dream

##### KS 3 Short Challenge Level:

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

### Folding in Half

##### KS 4 Short Challenge Level:

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

### Right-angled Midpoints

##### KS 4 Short Challenge Level:

Weekly Problem 33 - 2017
If the midpoints of the sides of a right angled triangle are joined, what is the perimeter of this new triangle?

### Right Angled Possibilities

##### KS 4 Short Challenge Level:

Weekly Problem 23 - 2007
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?

##### KS 4 Short Challenge Level:

Weekly Problem 40 - 2009
This quadrilateral has an unusual shape. Are you able to find its area?

### Rectangle Rearrangement

##### KS 4 Short Challenge Level:

Weekly Problem 48 - 2007
A 3x8 rectangle is cut into two pieces... then rearranged to form a right-angled triangle. What is the perimeter of the triangle formed?

### Tetromino Diagonal

##### KS 4 Short Challenge Level:

Weekly Problem 10 - 2010
Can you calculate the length of this diagonal line?

### Out of the Window

##### KS 4 Short Challenge Level:

Weekly Problem 3 - 2012
Find out how many pieces of hardboard of differing sizes can fit through a rectangular window.

### Crane Arm

##### KS 4 Short Challenge Level:

Weekly Problem 21 - 2014
A parallelogram is formed by joining together four equilateral triangles. What is the length of the longest diagonal?

### Folded Over

##### KS 4 Short Challenge Level:

Weekly Problem 11 - 2012
A rectangular piece of paper is folded. Can you work out one of the lengths in the diagram?

### Walk the Plank

##### KS 4 Short Challenge Level:

Weekly Problem 22 - 2006
A rectangular plank fits neatly inside a square frame when placed diagonally. What is the length of the plank?

### Unusual Polygon

##### KS 4 Short Challenge Level:

Weekly Problem 20 - 2011
What is the perimeter of this unusually shaped polygon...

### Winding Vine

##### KS 4 Short Challenge Level:

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

### Common Tangent

##### KS 4 Short Challenge Level:

Weekly Problem 33 - 2007
Two circles touch, what is the length of the line that is a tangent to both circles?

### Indigo Interior

##### KS 4 Short Challenge Level:

Weekly Problem 5 - 2013
The diagram shows 8 shaded squares inside a circle. What is the shaded area?

### Triangular Teaser

##### KS 4 Short Challenge Level:

Weekly Problem 14 - 2014
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?

### Semicircle in a Semicircle

##### KS 4 Short Challenge Level:

Weekly Problem 2 - 2008
The diagram shows two semicircular arcs... What is the diameter of the shaded region?

### Circle Time

##### KS 4 Short Challenge Level:

Weekly Problem 19 - 2010
Three circles of different radii each touch the other two. What can you deduce about the arc length between these points?

### Snapped Palm Tree

##### KS 4 Short Challenge Level:

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

### Interior Squares

##### KS 4 Short Challenge Level:

Weekly Problem 5 - 2008
Calculate the ratio of areas of these squares which are inscribed inside a semi-circle and a circle.

### Centre Square

##### KS 4 Short Challenge Level:

Weekly Problem 3 - 2011
What does Pythagoras' Theorem tell you about the radius of these circles?

### Oh So Circular

##### KS 4 Short Challenge Level:

Weekly Problem 24 - 2008
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?

### Salt's Mill

##### KS 4 Short Challenge Level:

Weekly Problem 49 - 2015
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?

### Hexagon Perimeter

##### KS 4 Short Challenge Level:

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

### Ice Cream Tangent

##### KS 4 Short Challenge Level:

Weekly Problem 28 - 2008
The diagram shows a semi-circle and an isosceles triangle which have equal areas. What is the value of tan x?

### Distance to the Corner

##### KS 4 Short 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?

### When the Boat Comes In

##### KS 4 Short 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?

### Symmetric Angles

##### KS 4 Short Challenge Level:

Weekly Problem 43 - 2009
This diagram has symmetry of order four. Can you use different geometric properties to find a particular length?

### Height of the Tower

##### KS 4 Short Challenge Level:

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

### Overlapping Ribbons

##### KS 4 Short Challenge Level:

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

### One or Two

##### KS 4 Short Challenge Level:

Weekly Problem 16 - 2014
The diagrams show squares placed inside semicircles. What is the ratio of the shaded areas?

### The Roller and the Triangle

##### KS 4 Short Challenge Level:

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