Geometry

Geometry is the branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogues. Expected questions involve calculating lengths, angles, areas, and volumes of various shapes, understanding geometric theorems, and solving problems related to spatial reasoning.

Solid Geometry / Geometry in Space Trigonometry Spherical Geometry Plane Geometry Vectors

• THE GARDEN PUZZLE

  Professor Rackbrain tells me that he was recently smoking a friendly pipe under a tree in the garden of a country acquaintance. The garden was enclosed by four straight walls, and his friend informed him that he had measured these and found the lengths to be `80, 45, 100`, and `63` yards respectively. "Then," said the professor, "we can calculate the exact area of the garden." "Impossible," his host replied, "because you can get an infinite number of different shapes with those four sides." "But you forget," Rackbrane said, with a twinkle in his eye, "that you told me once you had planted this tree equidistant from all the four corners of the garden." Can you work out the garden's area?
  Topics:

  Geometry -> Area Calculation Geometry -> Plane Geometry -> Circles Algebra -> Equations Algebra -> Word Problems
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 182

• ST. GEORGE'S BANNER

  At a celebration of the national festival of St. George's Day I was contemplating the familiar banner of the patron saint of our country. We all know the red cross on a white ground, shown in our illustration. This is the banner of St. George. The banner of St. Andrew (Scotland) is a white "St. Andrew's Cross" on a blue ground. That of St. Patrick (Ireland) is a similar cross in red on a white ground. These three are united in one to form our Union Jack.

  Now on looking at St. George's banner it occurred to me that the following question would make a simple but pretty little puzzle. Supposing the flag measures four feet by three feet, how wide must the arm of the cross be if it is required that there shall be used just the same quantity of red and of white bunting?

  

  Topics:

  Geometry -> Area Calculation Algebra -> Word Problems
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 185

• THE YORKSHIRE ESTATES

   

  I was on a visit to one of the large towns of Yorkshire. While walking to the railway station on the day of my departure a man thrust a hand-bill upon me, and I took this into the railway carriage and read it at my leisure. It informed me that three Yorkshire neighbouring estates were to be offered for sale. Each estate was square in shape, and they joined one another at their corners, just as shown in the diagram. Estate A contains exactly `370` acres, B contains `116` acres, and C `74` acres.

  Now, the little triangular bit of land enclosed by the three square estates was not offered for sale, and, for no reason in particular, I became curious as to the area of that piece. How many acres did it contain?

  Topics:

  Geometry -> Area Calculation Geometry -> Plane Geometry -> Triangles Geometry -> Plane Geometry -> Pythagorean Theorem
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 189

• FARMER WURZEL'S ESTATE

   

  I will now present another land problem. The demonstration of the answer that I shall give will, I think, be found both interesting and easy of comprehension.

  Farmer Wurzel owned the three square fields shown in the annexed plan, containing respectively `18, 20`, and `26` acres. In order to get a ring-fence round his property he bought the four intervening triangular fields. The puzzle is to discover what was then the whole area of his estate.

  Topics:

  Geometry -> Area Calculation Geometry -> Plane Geometry -> Triangles Geometry -> Plane Geometry -> Pythagorean Theorem
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 190

• THE SHEEP-FOLD

  It is a curious fact that the answers always given to some of the best-known puzzles that appear in every little book of fireside recreations that has been published for the last fifty or a hundred years are either quite unsatisfactory or clearly wrong. Yet nobody ever seems to detect their faults. Here is an example:—A farmer had a pen made of fifty hurdles, capable of holding a hundred sheep only. Supposing he wanted to make it sufficiently large to hold double that number, how many additional hurdles must he have?
  Topics:

  Geometry -> Area Calculation Logic -> Reasoning / Logic
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 193

• THE GARDEN WALLS

   

  A speculative country builder has a circular field, on which he has erected four cottages, as shown in the illustration. The field is surrounded by a brick wall, and the owner undertook to put up three other brick walls, so that the neighbours should not be overlooked by each other, but the four tenants insist that there shall be no favouritism, and that each shall have exactly the same length of wall space for his wall fruit trees. The puzzle is to show how the three walls may be built so that each tenant shall have the same area of ground, and precisely the same length of wall.

  Of course, each garden must be entirely enclosed by its walls, and it must be possible to prove that each garden has exactly the same length of wall. If the puzzle is properly solved no figures are necessary.

  Topics:

  Geometry -> Area Calculation Geometry -> Plane Geometry -> Circles Combinatorics -> Combinatorial Geometry -> Cut a Shape / Dissection Problems
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 194

• THE TETHERED GOAT

  Here is a little problem that everybody should know how to solve. The goat is placed in a half-acre meadow, that is in shape an equilateral triangle. It is tethered to a post at one corner of the field. What should be the length of the tether (to the nearest inch) in order that the goat shall be able to eat just half the grass in the field? It is assumed that the goat can feed to the end of the tether.
  Topics:

  Geometry -> Area Calculation Geometry -> Plane Geometry -> Triangles Geometry -> Plane Geometry -> Angle Calculation
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 196

• A NEW MATCH PUZZLE

  In the illustration eighteen matches are shown arranged so that they enclose two spaces, one just twice as large as the other. Can you rearrange them (`1`) so as to enclose two four-sided spaces, one exactly three times as large as the other, and (`2`) so as to enclose two five-sided spaces, one exactly three times as large as the other? All the eighteen matches must be fairly used in each case; the two spaces must be quite detached, and there must be no loose ends or duplicated matches.
  Topics:

  Geometry -> Area Calculation Combinatorics -> Combinatorial Geometry -> Cut a Shape / Dissection Problems Puzzles and Rebuses -> Matchstick Puzzles
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 204

• COUNTING THE RECTANGLES

  Can you say correctly just how many squares and other rectangles the chessboard contains? In other words, in how great a number of different ways is it possible to indicate a square or other rectangle enclosed by lines that separate the squares of the board?
  Topics:

  Combinatorics Geometry -> Area Calculation Number Theory -> Triangular Numbers
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 347

• Question

  There is a billiard table in the shape of a triangle whose angles are equal to \(90^{\circ}\), \(30^{\circ}\) and \(60^{\circ}\). 

   

  Given a right triangle shaped billiard table, with "pockets" in its corners. One of its acute angles is \(30^{\circ}\). From this corner (the thirty-degree angle) a ball is launched towards the midpoint of the opposite side of the triangle (the median). Prove that if the ball is reflected more than eight times (angle of incidence equals angle of reflection), then eventually the ball will enter the "pocket" located at the 60-degree corner of the triangle.

  Topics:

  Geometry -> Plane Geometry -> Triangles Geometry -> Plane Geometry -> Angle Calculation Geometry -> Plane Geometry -> Plane Transformations -> Congruence Transformations (Isometries) -> Reflection