The Apothem The apothem (a) is the segment drawn from the center of the polygon to the midpoint of the side (and perpendicular to the side)

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The Apothem The apothem (a) is the segment drawn from the center of the polygon to the midpoint of the side (and perpendicular to the side)

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Deriving the Formula – Squares The diagonals of the square divide it into four triangles with base s and height a. The area of each triangle is sa. Since there are 4 triangles, the total area is 4( )sa or (4s)a. Since the perimeter (p) = 4s the formula becomes A = ap

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Deriving the Formula – Triangles Connecting the center of the equilateral triangle to each vertex creates three congruent triangles with area A = sa. Since there are three triangles, the total area is 3( )sa, or (3s)a. Since the perimeter = 3s, the formula may be written A = ap

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Deriving the Formula – Regular Hexagons Connecting the center of the regular hexagon to each vertex creates six congruent triangles with area A = sa. Since there are six triangles, the total area is 6( )sa, or (6s)a. Since the perimeter = 6s, the formula may be written A = ap

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Finding the apothem – Square The apothem of a square is one-half the length of the side. If s = 15, a = ? a = 7.5 If a = 14, s = ? s = 28

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Find the apothem – Triangles The apothem of an equilateral triangle is the short leg of a 30- 60-90 triangle where s/2 is the long leg. 30 60 90 Then a = (s/2)/ 3 or

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Find the apothem – Triangles If s = 18, a = ?If s = 24, a = ?If s = 10, a = ?

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Find the side – Triangles If the apothem is 6 cm, the side = ? If the apothem is 2.5 cm, the side = ?

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Finding the apothem – Hexagons The apothem of a regular hexagon is the long leg of a 30-60-90 triangle. 60 90 30 Therefore, the apothem is (s/2)

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Finding the apothem – hexagons If the side = 12 the apothem = ? If the side = 5 the apothem = ?

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Finding the side – hexagons If the apothem = 12, the side = ? If the apothem = 16, the side = ?

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Finding the area – Squares a = 6 cm Find the area A = 144 cm 2 A = 288 cm 2 A = 50 cm 2

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Finding the Area – Triangles If a = 3 cm, find the area of the triangle

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Finding the Area – Triangles If a = 5 cm, find the area of the triangle

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Finding the Area – Triangles

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If the side of the triangle = 10 cm, find the area of the triangle

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Finding the Area – Triangles

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Finding the area – hexagons If the a = 6 cm, find the area of the hexagon.

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Finding the area – hexagons

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