# Pythagoras' Theorem & Trigonometry

This is part of ourĀ Secondary Curriculum collection of favourite rich tasks arranged by topic.

### Garden Shed

##### Stage: 3 Challenge Level:

Can you minimise the amount of wood needed to build the roof of my garden shed?

### Where Is the Dot?

##### Stage: 4 Challenge Level:

A dot starts at the point (1,0) and turns anticlockwise. Can you estimate the height of the dot after it has turned through 45 degrees? Can you calculate its height?

### Compare Areas

##### Stage: 4 Challenge Level:

Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?

### The Spider and the Fly

##### Stage: 4 Challenge Level:

A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?

### Semi-detached

##### Stage: 4 Challenge Level:

A square of area 40 square cms is inscribed in a semicircle. Find the area of the square that could be inscribed in a circle of the same radius.

### Where to Land

##### Stage: 4 Challenge Level:

Chris is enjoying a swim but needs to get back for lunch. If she can swim at 3 m/s and run at 7m/sec, how far along the bank should she land in order to get back as quickly as possible?

### Trigonometric Protractor

##### Stage: 4 Challenge Level:

An environment that simulates a protractor carrying a right- angled triangle of unit hypotenuse.

### Inscribed in a Circle

##### Stage: 4 Challenge Level:

The area of a square inscribed in a circle with a unit radius is, satisfyingly, 2. What is the area of a regular hexagon inscribed in a circle with a unit radius?

### Three Cubes

##### Stage: 4 Challenge Level:

Can you work out the dimensions of the three cubes?

### Hexy-metry

##### Stage: 4 Challenge Level:

A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?

### Ladder and Cube

##### Stage: 4 Challenge Level:

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

##### Stage: 4 Challenge Level:

Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.

### Far Horizon

##### Stage: 4 Challenge Level:

An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?

### Three by One

##### Stage: 5 Challenge Level:

There are many different methods to solve this geometrical problem - how many can you find?

### Cubestick

##### Stage: 5 Challenge Level:

Stick some cubes together to make a cuboid. Find two of the angles by as many different methods as you can devise.

### Pythagoras' Theorem and Trigonometry - Short Problems

##### Stage: 3 and 4

A collection of short problems on Pythagoras's Theorem and Trigonometry.