Pythagoras' Theorem - Short Problems


This is part of our collection of Short Problems.

You may also be interested in our longer problems on Pythagoras' Theorem.

Printable worksheets containing selections of these problems are available here:

Stage 4 ★ Sheet 1 Solutions            Stage 4 ★★ Sheet 1 Solutions
          Sheet 2 Solutions

Right-angled Midpoints

KS 3 & 4 Short Challenge Level:

Challenge Level:1

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?

Semicircle in a Semicircle

KS 3 & 4 Short Challenge Level:

Challenge Level:2 Challenge Level:2

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

Rectangle Rearrangement

KS 4 Short Challenge Level:

Challenge Level:1

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?

Crane Arm

KS 4 Short Challenge Level:

Challenge Level:1

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

Out of the Window

KS 4 Short Challenge Level:

Challenge Level:1

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

Folding in Half

KS 4 Short Challenge Level:

Challenge Level:1

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

Folded Over

KS 4 Short Challenge Level:

Challenge Level:1

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

Unusual Quadrilateral

KS 4 Short Challenge Level:

Challenge Level:1

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

Right Angled Possibilities

KS 4 Short Challenge Level:

Challenge Level:1

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?

Tetromino Diagonal

KS 4 Short Challenge Level:

Challenge Level:1

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

Centre Square

KS 4 Short Challenge Level:

Challenge Level:2 Challenge Level:2

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

Interior Squares

KS 4 Short Challenge Level:

Challenge Level:2 Challenge Level:2

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

Unusual Polygon

KS 4 Short Challenge Level:

Challenge Level:2 Challenge Level:2

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

Walk the Plank

KS 4 Short Challenge Level:

Challenge Level:2 Challenge Level:2

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

Indigo Interior

KS 4 Short Challenge Level:

Challenge Level:2 Challenge Level:2

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:

Challenge Level:2 Challenge Level:2

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?

Common Tangent

KS 4 Short Challenge Level:

Challenge Level:2 Challenge Level:2

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

Circle Time

KS 4 Short Challenge Level:

Challenge Level:2 Challenge Level:2

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?

Salt's Mill

KS 4 Short Challenge Level:

Challenge Level:3 Challenge Level:3 Challenge Level:3

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?

Oh So Circular

KS 4 Short Challenge Level:

Challenge Level:3 Challenge Level:3 Challenge Level:3

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?

One or Two

KS 4 Short Challenge Level:

Challenge Level:3 Challenge Level:3 Challenge Level:3

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