List

Pythagoras' Theorem and Trigonometry - Short Problems

Small pepper seedlings in turquoise pots.
problem

Crane Arm

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
problem

Right-Angled Midpoints

Age
14 to 16
Challenge level

If the midpoints of the sides of a right angled triangle are joined, what is the perimeter of this new triangle?

Small pepper seedlings in turquoise pots.
problem

Pythagoras' Dream

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
problem

Out of the Window

Age
14 to 16
Challenge level

Find out how many pieces of hardboard of differing sizes can fit through a rectangular window.

Small pepper seedlings in turquoise pots.
problem

Right Angled Possibilities

Age
14 to 16
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?

Small pepper seedlings in turquoise pots.
problem

Rectangle Rearrangement

Age
14 to 16
Challenge level

A 3×8 rectangle is cut into two pieces... then rearranged to form a right-angled triangle. What is the perimeter of the triangle formed?

Small pepper seedlings in turquoise pots.
problem

Tetromino Diagonal

Age
14 to 16
Challenge level

Can you calculate the length of this diagonal line?

Small pepper seedlings in turquoise pots.
problem

Folding in Half

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
problem

Unusual Polygon

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
problem

Triangular Teaser

Age
14 to 16
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?

Small pepper seedlings in turquoise pots.
problem

Winding Vine

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
problem

Question of Three Sides

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
problem

Building Blocks

Age
14 to 16
Challenge level

Can you find the length of AB in this diagram?

Small pepper seedlings in turquoise pots.
problem

Snapped Palm Tree

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
problem

Folded Over

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
problem

Hexagon Perimeter

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
problem

Walk the Plank

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
problem

Arc Radius

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
problem

Unusual Quadrilateral

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
problem

Symmetric Angles

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
problem

Strike a Chord

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
problem

Diagonal Area

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
problem

Three Right Angles

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
problem

Indigo Interior

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
problem

One or Two

Age
14 to 16
Challenge level

The diagrams show squares placed inside semicircles. What is the ratio of the shaded areas?

Small pepper seedlings in turquoise pots.
problem

Salt's Mill

Age
14 to 16
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?

Small pepper seedlings in turquoise pots.
problem

Triple Pythagoras

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
problem

Integers on a Sphere

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
problem

Overlapping Ribbons

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
problem

The Roller and the Triangle

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
problem

Four Circles

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
problem

Smartphone Screen

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
problem

Distance to the Corner

Age
14 to 16
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?

Small pepper seedlings in turquoise pots.
problem

Common Tangent

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
problem

Folded Rectangle

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
problem

Triangle Radius

Age
14 to 16
Challenge level

Can you find the radii of the small circles?

Small pepper seedlings in turquoise pots.
problem

Semicircle in a Semicircle

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
problem

When the Boat Comes In

Age
14 to 16
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?

Small pepper seedlings in turquoise pots.
problem

Interior Squares

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
problem

Height of the Tower

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
problem

Oh so Circular

Age
14 to 16
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?

Small pepper seedlings in turquoise pots.
problem

Diamond Ring

Age
14 to 16
Challenge level

Find the radius of the stone in this ring.

Small pepper seedlings in turquoise pots.
problem

Ice Cream Tangent

Age
14 to 16
Challenge level

The diagram shows a semi-circle and an isosceles triangle which have equal areas. What is the value of tan x?

Small pepper seedlings in turquoise pots.
problem

Square Overlap

Age
14 to 16
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?

Small pepper seedlings in turquoise pots.
problem

Circle Time

Age
14 to 16
Challenge level

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

Small pepper seedlings in turquoise pots.
problem

Centre Square

Age
14 to 16
Challenge level

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