AREAS OF POLYGONS

Excerpts

• AREAS OF POLYGONS

  The **area of a surface** is the *number of units of surface* it contains.

• AREAS OF POLYGONS

  Two rectangles having equal altitudes are to each other as their bases.

• AREAS OF POLYGONS

  The **unit of surface** is a square whose side is a *unit of length*.

• AREAS OF POLYGONS

  Plane figures that *have equal areas but cannot be made to coincide* are called **equivalent**.

• AREAS OF POLYGONS

  In propositions relating to *areas*, the words “rectangle,” “triangle,” etc., are often used for “area of rectangle,” “area of triangle,” etc.

• AREAS OF POLYGONS

  The area of a rectangle is equal to the product of its base by its altitude.

• AREAS OF POLYGONS

  When the base and altitude each contain the linear unit an integral number of times, this proposition is rendered evident by dividing the figure into squares, each equal to the unit of surface.

• AREAS OF POLYGONS

  The area of an irregular polygon may be found by dividing the polygon into triangles, and by finding the area of each of these triangles separately.

• AREAS OF POLYGONS

  The square on the hypotenuse of a right triangle is equivalent to the sum of the squares on the two legs.

Equations

• AREAS OF POLYGONS

  \dfrac{\rect AF}{\rect AC} = \dfrac{AE}{AB}

  Two rectangles on the same altitude are to each other as their bases, so the ratio of the rectangles equals the ratio of the bases AE and AB.

• AREAS OF POLYGONS

  \dfrac{R}{U} = \dfrac{a × b}{1 × 1}

  The ratio of a rectangle R to the unit of surface U equals the product of its altitude and base, each measured in linear units.

• AREAS OF POLYGONS

  \Par AEFD = a × b

  The area of parallelogram AEFD equals its base b times its altitude a.

• AREAS OF POLYGONS

  \triangle{}ABC=\frac{1}{2}a × b

  The area of a triangle is half the product of its base by its altitude.

• AREAS OF POLYGONS

  ABCH=\frac{1}{2}a(b+b')

  The area of trapezoid ABCH equals half its altitude times the sum of its two bases.

• AREAS OF POLYGONS

  \dfrac{\triangle ABC} {\triangle ADE} = \dfrac{AB × AC} {AD × AE}

  Two triangles with one equal angle at A are to each other as the products of the two sides enclosing that angle.

• AREAS OF POLYGONS

  S:S'=\overline{AB}^2:\overline{A'B'^2}

  The areas of two similar polygons are to each other as the squares of any two homologous sides. (The book's typesetting places the square exponent inside the second overline; the intended relation is the squares of AB and A'B'.)

• AREAS OF POLYGONS

  BE \Bumpeq CH + AF

  The square on the hypotenuse of a right triangle is equivalent to the sum of the squares on the two legs.

• AREAS OF POLYGONS

  \overline{NP}^2 = MN × NO = a × b

  The square constructed on NP equals the product of the segments MN (= a) and NO (= b) of the diameter, so the square is equivalent to the parallelogram of base b and altitude a.

• AREAS OF POLYGONS

  \overline{BD}^2 + \overline{AC}^2 = \overline{AB}^2 + \overline{DC}^2

  For a right triangle ABC with right angle at C and a line BD cutting AC at D, the sum of the squares on BD and AC equals the sum of the squares on AB and DC.

• AREAS OF POLYGONS

  = \sqrt{s(s - a)(s - b)(s - c)}

  The area of a triangle with sides a, b, c equals the square root of s(s-a)(s-b)(s-c), where s is half the perimeter.

• AREAS OF POLYGONS

  = \dfrac{abc}{4R}

  The area of a triangle equals the product of its three sides divided by four times the radius of its circumscribed circle.

• AREAS OF POLYGONS

  \dfrac{a}{2} × \dfrac{a\sqrt{3}}{2} = \dfrac{a^2\sqrt{3}}{4}

  The area of an equilateral triangle of side a is a squared times the square root of 3, divided by 4.

Problems