Pythagoras' Theorem and Trigonometry - Short Problems
Out of the Window
Find out how many pieces of hardboard of differing sizes can fit through a rectangular window.
Crane Arm
A parallelogram is formed by joining together four equilateral triangles. What is the length of the longest diagonal?
Right-angled Midpoints
If the midpoints of the sides of a right angled triangle are joined, what is the perimeter of this new triangle?
Right Angled Possibilities
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?
Rectangle Rearrangement
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
Can you calculate the length of this diagonal line?
Pythagoras' Dream
Can you work out the area of this isosceles right angled triangle?
Three Right Angles
Work your way through these right-angled triangles to find $x$.
Folded Over
A rectangular piece of paper is folded. Can you work out one of the lengths in the diagram?
Hexagon Perimeter
A circle of radius 1 is inscribed in a regular hexagon. What is the perimeter of the hexagon?
Walk the Plank
A rectangular plank fits neatly inside a square frame when placed diagonally. What is the length of the plank?
Folding in Half
How does the perimeter change when we fold this isosceles triangle in half?
Arc Radius
Two arcs are drawn in a right-angled triangle as shown. What is the length $r$?
Unusual Polygon
What is the perimeter of this unusually shaped polygon...
Unusual Quadrilateral
This quadrilateral has an unusual shape. Are you able to find its area?
Winding Vine
A vine is growing up a pole. Can you find its length?
Snapped Palm Tree
A palm tree has snapped in a storm. What is the height of the piece that is still standing?
Symmetric Angles
This diagram has symmetry of order four. Can you use different geometric properties to find a particular length?
Strike a Chord
Can you work out the radius of a circle from some information about a chord?
Question of Three Sides
Can you find the length of the third side of this triangle?
Triangular Teaser
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
A square has area 72 cm$^2$. Find the length of its diagonal.
One or Two
The diagrams show squares placed inside semicircles. What is the ratio of the shaded areas?
Four Circles
Can you find the radius of the larger circle in the diagram?
Circle Time
Three circles of different radii each touch the other two. What can you deduce about the arc length between these points?
Triple Pythagoras
Can you work out the length of the diagonal of the cuboid?
Interior Squares
Calculate the ratio of areas of these squares which are inscribed inside a semi-circle and a circle.
Height of the Tower
How do these measurements enable you to find the height of this tower?
Centre Square
What does Pythagoras' Theorem tell you about the radius of these circles?
Integers on a Sphere
Can you find all the integer coordinates on a sphere of radius 3?
Oh So Circular
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?
Salt's Mill
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
Can you find the length and width of the screen of this smartphone in inches?
Ice Cream Tangent
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
Can you find the distance from the well to the fourth corner, given the distance from the well to the first three corners?
Overlapping Ribbons
Two ribbons are laid over each other so that they cross. Can you find the area of the overlap?
Square Overlap
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
Two circles touch, what is the length of the line that is a tangent to both circles?
Folded Rectangle
Can you find the perimeter of the pentagon formed when this rectangle of paper is folded?
Indigo Interior
The diagram shows 8 shaded squares inside a circle. What is the shaded area?
The Roller and the Triangle
How much of the inside of this triangular prism can Clare paint using a cylindrical roller?
Semicircle in a Semicircle
The diagram shows two semicircular arcs... What is the diameter of the shaded region?
When the Boat Comes In
When you pull a boat in using a rope, does the boat move more quickly, more slowly, or at the same speed as you?
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