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Description

permacharts TM
Geometry
Geometry
QUADRILATERALS CIRCLES & CONICS
GENERAL QUADRILATERALS A d Note: For the following figures, C = circumference and
D s = arc length
Sum of the Interior Angles
A + B + C + D = 360°
a θ CIRCLES
Diameter
Perimeter a + b + c + d p
c D = 2R
B q s
Area Circumference
1 1 segment
pqsinθ =−θ+b2 2d2 2 a c tan b 2πR = πD h
2 4 a
1 2 Area
− − = + − 4pq2 2 2b22d a c C b ▯
4 πRπD= 1 θ R
4 sector
Theorem: Diagonals of a quadrilateral with
consecutive sides a, b, c, 2nd 2 ar2 2 Sector & Segment of Circle
perpendicular if and only if ac = b + d b a
θ = 2c2s−1 = sin
RECTANGLES R 2R
D b C sR= θ
Interior Angles hR=b−
A = B = C = D = 90° θ
a = 2 sin = 2 tan
p 2 2
Perimeter a a 1 2 2
2(a + b) bR= cos = R c4
2 2
Area (sector) = Rs =1 R2θ
Area 2 2
ab A b B 1 2
Area (segment) = 2R ( )− sin
TRAPEZOIDS
Sum of the Interior Angles D a C PARABOLAS (SEGMENT)
A + B + C + D = 360°, a||b
Arc Length
Perimeter h 1 2 2 b26 a + + a B
2 b a+ +6 8aln b
ab+h + + 1 1
siθ β sin
θ β Area
Area A b B a
2
1 3 ab
2h()+b
PARALLELOGRAMS 1 1
Interior Angles D b A 2 b 2b C
C ELLIPSES
A = C, B = D Diameter
A + B = 180° q
2a = major axis
2b = minor axis
Interior Lengths
ha A a B sin a a
a
Area
2 2 p πab
p a bb aA 2 cos
2 2 h
q abb B 2 cos Perimeter b
A b B ab−2
Perimeter k = a
2(a + b)
π
2 2 2
Area P = 4 ∫ d sin θθ
bh = absinA = absinB 0
= 2π 1ab2 2 (approximately)
2
1 GEOMETRY • A-588-6 w w w . p e r m a c h a r t s . c o m © 1997-2012 Mindsource Technologies Inc. permachartsM
TRIANGLES
• For the following triangles, s = semiperimeter2(a + b + c), RIGHT TRIANGLES B
r = radius of inscribed circle, R = radius of circumscribed circle
Interior Angles Area
A + B = C = 90° 1 1 c
GENERAL TRIANGLES ab = hc
Sum of the Interior Angles Radii 2 2
Radii Perimeter h a
A + B + C = 180° ab sinC C ab a + b + c
r= = () tan r= A
Height cs2 abc ab c
R = = R c1 b C
h = asinB = bsinA 2sinC K C 2
Perimeter EQUILATERAL TRIANGLES C
a + b + c
Interior Angles Perimeter
b a A = B = C = 60° 3a
Area
1 1 abc h
CcKah = = sin Radii Area a a
2 2 4 R h
==rs −s() () ()−s c − r=a1 3 h a a 3 = 1
6 4 2
1
A c B R=a 3 A a B
3
PLANE AREA APPROXIMATION FORMULAS REGULAR POLYGONS
• Divide the planar area ABCD into n parallel strips of equal • For the following polygons, n = number of sides, r = radius of
thickness h inscribed circle, and R = radius of circumscribed circle
• Let 0 ,1x 2 x , …n x denote the lengths of the vertical cords
C B
• h = B− Xn
n a R
2n r
h
θ
X2
Interior Angles Perimeter
n −2 na = 2nRsin180°
X1 θ = n 180°,>n 2 n
D X A Length Radii Area
0 180 ° 1 180° 1 180°
a = 2 tan r = cot na cot
Trapezoidal Rule n 2 n 4 n
1 1 2 180 °
Area+=+hx x0 1 2 x 1 n−n = 2Rsin180 ° R = 1 csc180° = nr tan
2 2 n 2 n n
1 2 360°
Simpson’s Rule (n even) = 2nR sin n
1
Area =++h(+0 1 2 3 42 x 22x 14 x n−nn x
3 n Polygon r R Area
2
Durand’s Rule 3 Triangle 0.2887a 0.5774a 0.4330a
4 Square 0.5000a 0.7071a 1.0000a 2
4 11 11 4 2
A+r+=+h+10 x02110 x2 1x x n−n− n10 10 5 Pentagon 0.6882a 0.8506a 1.7205a
6 Hexagon 0.8660a 1.0000a 2.5981a 2
7 Heptagon 1.0383a 1.1524a 3.6339a 2
Weddle’s Rule (n = 6) 2
3 8 Octagon 1.2071a 1.3066a 4.8284a
+rea+=++ hx(0 12 34 56 x x x x 9 Nonagon 1.3737a 1.4619a 6.1818a 2
10 10 Decagon 1.5388a 1.6180a 7.6942a 2
2 GEOMETRY • A-588-6 w w w.permacharts.com © 1997-2012 Mindsource Technologies Inc. permachartsM
PLANE SOLIDS
PARALLELEPIPED RIGHT

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