The Pythagorean Theorem In A Roof

Pythagorean formula c 2 a 2 b 2 where a b are the legs of a right triangle and c is the hypotenuse.

The pythagorean theorem in a roof. For example if a roof has a pitch of 4 12 then for every 12 inches the building extends horizontally it rises 4 inches. The greeks 1500 years ago used full scale drawings of the roofs to develop the rafter lengths and bevels and used the pythagorean theorem to square up the strings they used for the geometric layout of the roof. The square on the hypotenuse of a right triangle is equal to the sum of the squares on the two legs eves 80 81. The pythagorean theorem describes how the three sides of a right triangle are related in euclidean geometry.

The theorem can be proved algebraically using four copies of a right triangle with sides a a a b b b and c c c arranged inside a square with side c c c as in the top half of the diagram. In mathematics the pythagorean theorem also known as pythagoras s theorem is a fundamental relation in euclidean geometry among the three sides of a right triangle. The pythagorean theorem says that in a right triangle the square of a which is a a and is written a 2 plus the square of b b 2 is equal to the square of c c 2. Since the roof forms an isosceles triangle the perpendicular from the top of the roof must divide the across length in two equal halves.

This theorem is talking about the area of the squares that are built on each side of the right triangle. The triangles are similar with area 1 2 a b frac 1 2 ab 2 1 a b while the small square has side b a b a b a and area b a 2 b. Define points d and e on ab so that ad ae b. A 2 b 2 c 2 proof of the pythagorean theorem using algebra.

The pitch is commonly defined as the ratio of rise over run in the form of x 12. As usual ab c ac b bc a. The pythagorean theorem is just a special case of another deeper theorem from trigonometry called the law of cosines c 2 a 2 b 2 2 a b cos c where c is the angle opposite to the long side c. When c pi 2 or 90 degrees if you insist cos 90 0 and the term containing the cosine vanishes.

Proof 39 by j. Barry sutton the math gazette v 86 n 505 march 2002 p72 let in abc angle c 90 o. The pythagorean theorem is proven after two triangles are removed from each of the hexagons. Pythagorean theorem the theorem states that.

The rise is the height of the roof and the run is the horizontal span as pictured above. Let the length of one side of the roof be x ft. The roof pitch is the slope of the rafter. You can also think of this theorem as the hypotenuse formula.

It states that the sum of the squares of the sides of a right triangle equals the square of the hypotenuse.