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        
             

Right-angled Midpoints

Age 14 to 16 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

Age 14 to 16 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?

Unusual Quadrilateral

Age 14 to 16 Short Challenge Level:

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

Rectangle Rearrangement

Age 14 to 16 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

Age 14 to 16 Short Challenge Level:

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

Pythagoras' Dream

Age 14 to 16 Short Challenge Level:

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

Out of the Window

Age 14 to 16 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

Age 14 to 16 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

Age 14 to 16 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?

Folding in Half

Age 14 to 16 Short Challenge Level:

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

Unusual Polygon

Age 14 to 16 Short Challenge Level:

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

Square Overlap

Age 14 to 16 Short Challenge 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?

Winding Vine

Age 14 to 16 Short Challenge Level:

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

Common Tangent

Age 14 to 16 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

Age 14 to 16 Short Challenge Level:

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

Snapped Palm Tree

Age 14 to 16 Short Challenge Level:

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

Strike a Chord

Age 14 to 16 Short Challenge Level:

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

Question of Three Sides

Age 14 to 16 Short Challenge Level:

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

Triangular Teaser

Age 14 to 16 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?

Diagonal Area

Age 14 to 16 Short Challenge Level:

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

Semicircle in a Semicircle

Age 14 to 16 Short Challenge Level:

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

Circle Time

Age 14 to 16 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?

Three Right Angles

Age 14 to 16 Short Challenge Level:

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

Interior Squares

Age 14 to 16 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

Age 14 to 16 Short Challenge Level:

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

Diamond Ring

Age 14 to 16 Short Challenge Level:

Find the radius of the stone in this ring.

Walk the Plank

Age 14 to 16 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?

Building Blocks

Age 14 to 16 Short Challenge Level:

Can you find the length of AB in this diagram?

Arc Radius

Age 14 to 16 Short Challenge Level:

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

Distance to the Corner

Age 14 to 16 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?

Overlapping Ribbons

Age 14 to 16 Short Challenge Level:

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

Folded Rectangle

Age 14 to 16 Short Challenge Level:

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

Symmetric Angles

Age 14 to 16 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?

Triangle Radius

Age 14 to 16 Short Challenge Level:

Can you find the radii of the small circles?

The Roller and the Triangle

Age 14 to 16 Short Challenge Level:

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

When the Boat Comes In

Age 14 to 16 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?

One or Two

Age 14 to 16 Short Challenge Level:

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

Four Circles

Age 14 to 16 Short Challenge Level:

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

Triple Pythagoras

Age 14 to 16 Short Challenge Level:

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

Height of the Tower

Age 14 to 16 Short Challenge Level:

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

Hexagon Perimeter

Age 14 to 16 Short Challenge Level:

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

Integers on a Sphere

Age 14 to 16 Short Challenge Level:

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

Oh So Circular

Age 14 to 16 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

Age 14 to 16 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?

Smartphone Screen

Age 14 to 16 Short Challenge Level:

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

Ice Cream Tangent

Age 14 to 16 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?