10 1 geometry

PART 10.1: GEOMETRY X.1

PERIMETER:

Shape Equilateral Triangle Isosceles Triangle Scalene Triangle

Formula

Diagram

P = 3a P = 2a + b P = a+b+c

P = a + b + a2 + b2 P = 4a P = 2(a + b ) P = 4a

Right Triangle Square Rectangle Rhombus Equilateral Polygon

P = na

Regular Polygon

π  P = 2nb sin   n

General Polygon

P = ∑ ai

B is the distance between the center and a vertex

n

i =1

P = C = 2πr = πD

Circle

r12 + r22 2 P ≈ π 3(r1 + r2 ) − (3r1 + r2 )(r1 + 3r2 ) P ≈ 2π

(

Ellipse

∞  (2k )!2 e 2k  P = 2r1π 1 − ∑  k =1 2 k k 4 (2k − 1)    2 4 2 6   1  2  1× 3  e  1 × 3 × 5  e   − ... P = 2r1π 1 −  e −     2  2 × 3  3  2 × 3 × 6  5  

( )

e= Length of Edges around a Cube Length of Edges around a Cuboid

r1

P = 4(a + b + c ) s=

P 2

AREA: Shape

Triangle

r12 − r22

P = 12a

Semi Perimeter

X.2

)

Formula

A=

1 bh 2

Diagram

Triangle (SAS) Triangle (SAA) Triangle (Heron’s) Triangle (Equilateral)

Square Rectangle Parallelogram Rhombus Trapezium

Quadrilateral

Rectangle with rounded corners: Regular Pentagon

Regular Hexagon

Regular Heptagon

Regular Octagon

Regular Nonagon

Regular Decagon

Regular Undecagon

Regular Dodecagon

1 ab sin C 2 a 2 sin B sin C A= 2 sin A A = s (s − a )(s − b )(s − c ) A=

3 2 a 4 A = a2 A = ab A = bh = ab sin A A = a 2 sin A a+b A = h   s  A=

A=

(s − a )(s − b )(s − c )(s − d ) − k

 ∠AB + ∠CD  k = abcd × cos 2   2   d d sin I A= 1 2 2 A = lw − r 2 (4 − π ) A=

− 5 2a 2

2 5+ 5

(

5−3

)

3 3a 2 2 7a 2 A=  180  4 tan    7 

A=

2a 2 A= 2 −1 9a 2 A=  180  4 tan    9  A=

25 2a 2

(

5+ 5 3 5 −5 2

11a  180  4 tan    11  3a 2 A= 2− 3 A=

)

Regular 24-gon

Regular Polygon

Circle Cyclic Quadrilateral Area of a sector (degrees) Area of a sector (radians) Area of a segment (degrees)

Area of a segment (radians) Area of an annulus

Ellipse

X.3

6a 2 6 − 3+ 2 −2 na 2 A=  180  4 tan    n  P2 A=  180  4n tan    n   180  A = nr 2 tan    n  nR 2  360  A= sin  2  n  πd 2 Cd 2 A = πr = = 4 4 A = (s − a )(s − b )(s − c )(s − d ) Q A= × πr 2 360 1 A = r 2Q 2 r2  Q  A=  × π − sin Q  2  180  2 r A = (Q − sin Q ) 2 A=

(

A = π r2 − r1 A=

π 4

2

2

)

 w = π  2

r is radius of inscribed circle R is radius of circumscribed circle

2

lw = πr1 r2

VOLUME:

Shape Cube Cuboid Prism Pyramid Tetrahedron

Octahedron

Formula

V =a V = abc V = A(B ) × h 1 V = × A(B ) × h 3 2 V = × a3 12 2 V = × a3 3 3

Diagram

Dodecahedron

Icosahedron

V Cylinder

V

Sphere

V

Hemisphere Cap of a Sphere Cone

Doughnut

V

V

(

)

(

V = 2π 2 r2 r1 = 2

)

π2

4 2 πw  w  V= l −  4  3 4 V = πr1 r2 r3 3

Sausage

Ellipsoid

X.4

V

V

Ice-cream & Cone

15 + 7 5 × a3 4 53+ 5 = × a3 12 2 = πr h 4 = πr 3 3 2 3 = πr 3 1 2 = πh 3r1 + h 2 6 1 = πr 2 h 3 1 = πr 2 (h + 2r ) 3

V =

(b + a )(b − a )2

SURFACE AREA: Shape

Cube Cuboid Prism Regular Pyramid Tetrahedron

Formula

Diagram

SA = 6a 2 SA = 2(ab + bc + ca ) SA = 2 × A(B ) + P(B ) × a P (b ) × l SA = A(B ) + 2 2 SA = 3 × a

Octahedron

SA = 2 × 3 × a 2

Dodecahedron

SA = 3 × 25 + 10 5 × a 2

Icosahedron

SA = 5 × 3 × a 2 SA = 2πr (h + r )

Cylinder Sphere Hemisphere

SA = 4πr 2 SA = 3πr 2

(

)

Cone

SA = πr r + r 2 + h 2

Doughnut

SA = 4π r2 r1 = π b − a 2 SA = πwl

Sausage

2

2

(

2

L is the slant length

)

Ellipsoid

 a pb p + a p c p + b pc p   SA ≈ 4π ×  3   p ≈ 1.6075 p

X.5

DIAGONALS: Shape

Formula

Square

d = 2a

Rectangle

d = a 2 + b2 2a d= 5 −1 d = 2a 2a d= 2− 2 4a d= 5 −1

Regular Pentagon Regular Hexagon Regular Octagon

Regular Decagon Regular Dodecagon

Regular 24-gon

Even Sided Polygon

Odd Sided Polygon

Cuboid

X.5

d=

2 2a 3 −1 4a

8−2 6 −2 2 a d=  180  sin    n  a d=  90  2 sin    n 

d = a2 + b2 + c2 MISCELLANEOUS:

Euler’s Formula 3.5

d=

Faces + Verticies = Edges + 2

ABBREVIATIONS (3.1, 3.2, 3.3, 3.4)

A=area a=side ‘a’ b=base b=side ‘b’ C=circumference C=central angle c=side ‘c’ d=diameter

Diagram

d=diagonal d1=diagonal 1 d2=diagonal 2 E=external angle h=height I=internal angle l=length n=number of sides P=perimeter r=radius r1=radius 1 s=semi-perimeter SA=Surface Area V=Volume w=width