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

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?

problem

Tetromino Diagonal

Age

14 to 16

Challenge level

Can you calculate the length of this diagonal line?

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?

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?

problem

Pythagoras' Dream

Age

14 to 16

Challenge level

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

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.

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?

problem

Arc Radius

Age

14 to 16

Challenge level

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

problem

Unusual Quadrilateral

Age

14 to 16

Challenge level

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

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?

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?

problem

Diagonal Area

Age

14 to 16

Challenge level

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

problem

Three Right Angles

Age

14 to 16

Challenge level

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

problem

Folding in Half

Age

14 to 16

Challenge level

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

problem

Unusual Polygon

Age

14 to 16

Challenge level

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

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?

problem

Winding Vine

Age

14 to 16

Challenge level

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

problem

Question of Three Sides

Age

14 to 16

Challenge level

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

problem

Building Blocks

Age

14 to 16

Challenge level

Can you find the length of AB in this diagram?

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?

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?

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?

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?

problem

Triangle Radius

Age

14 to 16

Challenge level

Can you find the radii of the small circles?

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?

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?

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.

problem

Height of the Tower

Age

14 to 16

Challenge level

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

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?

problem

Diamond Ring

Age

14 to 16

Challenge level

Find the radius of the stone in this ring.

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?

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?

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?

problem

Centre Square

Age

14 to 16

Challenge level

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

problem

Indigo Interior

Age

14 to 16

Challenge level

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

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?

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?

problem

Triple Pythagoras

Age

14 to 16

Challenge level

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

problem

Integers on a Sphere

Age

14 to 16

Challenge level

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

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?

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?

problem

Four Circles

Age

14 to 16

Challenge level

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

problem

Smartphone Screen

Age

14 to 16

Challenge level

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

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?

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?

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?

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Pythagoras' Theorem and Trigonometry - Short Problems