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How was the formula for the area of a kite created?

2 Answers

Jan 5, 2016

Decompose a kite into two triangles and take the sum of the areas of the triangles

Explanation:

Consider the diagram below of a kite with diagonals of length $d$ $d$ and $w$ $w$ :
enter image source here

$¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ XXXXXXXXXXXXXXXXXXXXXXXXXXXXXX$ $bar(color(white)("XXXXXXXXXXXXXXXXXXXXXXXXXXXXXX")$

$△ A B D$ $triangle ABD$ has an area $= \frac{base \times height}{2}$ $=("base" xx "height")/2$

$XXXXXXXXXXXXX = \frac{w x}{2}$ $color(white)("XXXXXXXXXXXXX")=(wx)/2$

$¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ XXXXXXXXXXXXXXXXXXXXXXXXXXXXXX$ $bar(color(white)("XXXXXXXXXXXXXXXXXXXXXXXXXXXXXX")$

$△ B C D$ $triangle BCD$ has an area $= \frac{base \times height}{2}$ $=("base" xx "height")/2$

$XXXXXXXXXXXXX = \frac{w \times (d - x)}{2} = \frac{w d}{2} - \frac{w x}{2}$ $color(white)("XXXXXXXXXXXXX")=(w xx (d-x))/2 = (wd)/2-(wx)/2$

$¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ XXXXXXXXXXXXXXXXXXXXXXXXXXXXXX$ $bar(color(white)("XXXXXXXXXXXXXXXXXXXXXXXXXXXXXX")$

$Area of kite$ $"Area of kite"$
$XXX = A r e a_{ABD} + A r e a_{BCD}$ $color(white)("XXX")=Area_"ABD" + Area_"BCD"$

$XXX = (\frac{w x}{2}) + (\frac{w d}{2} - \frac{w x}{2})$ $color(white)("XXX")= ((wx)/2) + ((wd)/2 - (wx)/2)$

$XXX = \frac{w d}{2}$ $color(white)("XXX")=(wd)/2$

Jan 5, 2016

Here are a couple more images that might make the derivation of the formula for the area of a kite more intuitively obvious.

Explanation:

The kite deformed into a triangle:
enter image source here

The kite embedded in a rectangle:
enter image source here
Segments of the kite occupy $\frac{1}{2}$ $1/2$ of each quadrant of the rectangle (and thus has an area $= \frac{1}{2} \times$ $= 1/2 xx$ area of the rectangle).

Note that this second image implies that any convex quadrilateral with perpendicular diagonals (of which a kite is an example) has the same formula for its area.

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