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:

Challenge Level:2 Challenge Level:2

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

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.

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?

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?

Folding in Half

KS 4 Short Challenge Level:

Challenge Level:1

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

Right-angled Midpoints

KS 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?

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?

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?

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?

Tetromino Diagonal

KS 4 Short Challenge Level:

Challenge Level:1

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

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?

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.

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?

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?

Building Blocks

KS 4 Short Challenge Level:

Challenge Level:2 Challenge Level:2

Can you find the length of AB in this diagram?

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...

Winding Vine

KS 4 Short Challenge Level:

Challenge Level:2 Challenge Level:2

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

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?

Diamond Ring

KS 4 Short Challenge Level:

Challenge Level:2 Challenge Level:2

Find the radius of the stone in this ring.

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?

Snapped Palm Tree

KS 4 Short Challenge Level:

Challenge Level:2 Challenge Level:2

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

Question of Three Sides

KS 4 Short Challenge Level:

Challenge Level:2 Challenge Level:2

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

Square Overlap

KS 4 Short Challenge Level:

Challenge Level:2 Challenge Level:2

The top square has been rotated so that the squares meet at a 60$^\text{o}$ angle. What is the area of the overlap?

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?

Semicircle in a Semicircle

KS 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?

Hexagon Perimeter

KS 4 Short Challenge Level:

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

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

Triple Pythagoras

KS 4 Short Challenge Level:

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

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

Distance to the Corner

KS 4 Short Challenge Level:

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

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

Integers on a Sphere

KS 4 Short Challenge Level:

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

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

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?

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?

When the Boat Comes In

KS 4 Short Challenge Level:

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

When you pull a boat in using a rope, does the boat move more quickly, more slowly, or at the same speed as you?

Ice Cream Tangent

KS 4 Short Challenge Level:

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

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?

Height of the Tower

KS 4 Short Challenge Level:

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

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

Overlapping Ribbons

KS 4 Short Challenge Level:

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

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

Symmetric Angles

KS 4 Short Challenge Level:

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

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

The Roller and the Triangle

KS 4 Short Challenge Level:

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

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

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?