# 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

### Rectangle Rearrangement

##### Age 14 to 16 Short Challenge 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?

### Tetromino Diagonal

##### Age 14 to 16 Short Challenge Level:

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:

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:

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

### Right-angled Midpoints

##### Age 14 to 16 Short Challenge Level:

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:

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 Polygon

##### Age 14 to 16 Short Challenge Level:

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

##### Age 14 to 16 Short Challenge Level:

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

### Winding Vine

##### Age 14 to 16 Short Challenge Level:

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

### 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?

### Symmetric Angles

##### Age 14 to 16 Short Challenge Level:

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

### 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:

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.

### Three Right Angles

##### Age 14 to 16 Short Challenge Level:

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

### Folded Over

##### Age 14 to 16 Short Challenge Level:

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

### 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?

### Walk the Plank

##### Age 14 to 16 Short Challenge Level:

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

### Folding in Half

##### Age 14 to 16 Short Challenge Level:

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

### Building Blocks

##### Age 14 to 16 Short Challenge Level:

Can you find the length of AB in this diagram?

##### Age 14 to 16 Short Challenge Level:

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

### 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?

### 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?

### Common Tangent

##### Age 14 to 16 Short Challenge Level:

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

### 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?

### Indigo Interior

##### Age 14 to 16 Short Challenge Level:

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

##### 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?

### Semicircle in a Semicircle

##### Age 14 to 16 Short Challenge Level:

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

### 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:

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?

### Circle Time

##### Age 14 to 16 Short Challenge Level:

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

### Triple Pythagoras

##### Age 14 to 16 Short Challenge Level:

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

### Interior Squares

##### Age 14 to 16 Short Challenge Level:

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

### Height of the Tower

##### Age 14 to 16 Short Challenge Level:

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

### Centre Square

##### Age 14 to 16 Short Challenge Level:

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

### 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:

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?

### Diamond Ring

##### Age 14 to 16 Short Challenge Level:

Find the radius of the stone in this ring.

### Salt's Mill

##### Age 14 to 16 Short Challenge 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?

### 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:

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

##### 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?