DIMENSIONS OF PLANE FIGURES Machinery's Handbook, 31st Edition
77
Trapezoid:
A a b + ( ) h 2 ---------- = =
Area
a
Note: In Britain, this figure is called a trapezium and the figure below it is known as a trapezoid , which is the reverse of the US terms. Example: Side a = 23 meters, side b = 32 meters, and height h = 12 meters. Find the area. A a b + ( ) h 2 ---------- 23+32 ( ) 12 2 --------------- 55 12 × 2 --------- 330 m 2 = = = = × ----
h
b
Trapezium:
A H h + ( ) a bh cH + + 2 ------------------------ = =
Area
H
h
The area of a trapezium also can be found by dividing it into two triangles, as indicated by the dashed line. Each area is added to give the total area of the trapezium.
b
a
c
Example: Let a = 10 in., b = 2, c = 3 in., h = 8 in., and H = 12 in. Find the area. A H h + ( ) a bh cH + + 2 ------------------------ 12+8 ( ) 10 2 8 3 12 × + × + 2 ---------- ------------------------ = = 20 10 +16 +36 × 2 ---------------------- 252 2 ----- 126 in 2 = = = × ----- ( ) - ( ) ( ) --------- Regular Hexagon:
A = 2.598 s 2 = 2.598 R 2 = 3.464 r 2 R = s = radius of circumscribed circle = 1.155 r r = radius of inscribed circle = 0.866 s = 0.866 R s = R = 1.155 r Example: The side s of a regular hexagon is 40 millimeters. Find the area and the radius r of the inscribed (drawn inside) circle.
R
60°
r
120°
s
A 2.598 s 2 2.598 40 2 × 2.598 1600 × 4156.8mm 2 = = = = r 0.866 s 0.866 40 × 34.64mm = = =
Example: What is the length of the side of a hexagon circumscribed on (drawn around) a circle of 50 millimeters radius? In this case, because the hexagon is circumscribed on the circle, the circle is inscribed (drawn within) the hexagon. Hence, r = 50 mm and s 1.155 r 1.155 50 × 57.75mm = = =
Regular Octagon:
A = area = 4.828 s 2 = 2.828 R 2 = 3.314 r 2 R = radius of circumscribed circle = 1.307 s = 1.082 r r = radius of inscribed circle = 1.207 s = 0.924 R s = 0.765 R = 0.828 r Example: Find the area and the length of the side of an octagon inscribed (drawn inside) in a circle of 12 inches diameter. Diameter of circumscribed (drawn around) circle = 12 inches; hence, R = 6 in. A 2.828 R 2 2.828 6 2 × 2.828 36 × 101.81in 2 = = = = s 0.765 R 0.765 6 × 4.590in. = = =
R
45°
r
135°
s
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