Pythagoras' Theorem & Trigonometry

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

Tilted Squares

Age 11 to 14Challenge Level

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

Garden Shed

Age 11 to 14Challenge Level

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

Where Is the Dot?

Age 14 to 16Challenge 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

Age 14 to 16Challenge Level

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

The Spider and the Fly

Age 14 to 16Challenge 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

Age 14 to 16Challenge 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

Age 14 to 16Challenge Level

Chris is enjoying a swim but needs to get back for lunch. How far along the bank should she land?

Inscribed in a Circle

Age 14 to 16Challenge 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

Age 14 to 16Challenge Level

Can you work out the dimensions of the three cubes?

Hexy-metry

Age 14 to 16Challenge 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?

Age 14 to 16Challenge 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?

Age 14 to 16Challenge 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

Age 14 to 16Challenge 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

Age 16 to 18Challenge Level

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

Cubestick

Age 16 to 18Challenge 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

Age 11 to 16

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

You may also be interested in this collection of activities from the STEM Learning website, that complement the NRICH activities above.