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The document covers various geometric concepts including the Sine and Cosine rules, properties of triangles, quadrilaterals, and circles, as well as formulas for calculating areas and volumes of different shapes. It discusses theorems related to distances, slopes, and lines in geometry. Additionally, it includes information on mensuration and the properties of 3D shapes such as cubes, cylinders, and cones.
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Save Geometry Mensuration revision For Later Geometry. + (Plane, Zp ment
~*~ feo. Shtriandes aycin8 yl
A
* Sine Rule
‘ A c b
SinG= hh = bh = Csin&
c
Sines h > h=- bsinc ai ie .
Csink= beings,
co [QnA Tain Bg Grae ute | 4. 5 Qereroumemcnte:
iQicnaleeiag cae
& b Sard 2 (13-1)
ae Oo “&S "
t +t 4
Bb 248 i
*Cosind ule. Os
b
Cosas b+c> ;CosB= OHA ~b* iS
2bC 2ac
Costis; Q244% E B O ie
Bob
4 Byler t
* Pie s
s & GHEE Sal ee
BX Pi wilan
B D © OTA OPPS heC= se %
WM Acctie EL Cte Sut
PD Oh 4 CALS 4 (po*4 5e ce) rs
% Ot (ABC) = YE der BA fowmed by madiong. fe eS :
* Sn Anne, Median ee
Inievsea ot 90°. end ce ° & b c
SRC= AO + Act a
=
B & aqiscum mn dius e Tn dius sacra
Ro = aeeee ve A ole
es
© Porpendioulay + baso= 9 (4+) Bae
eDistancd Letuwéen inceatd and diseumoonht = [eset
Aron ob trhango = Ts (-Os 26-298 = C-27)8
tid Gz Semi-perimela —--RE_Cirevmradnus» \ ae
C= kypotenus m= Shechot medion, f i
T= tyradtus -
© AL amd CM ane _ +00 me dans >
14 ;
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°
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eyede
oD
. see
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" Otde Square Quoolet laters
© Peas BC (hth
% cdhabed 2 2
© Arg = a, sind.
AOiadipugseczake 70
® When dia
© 9% circle toucho of} edo g Ovech ebeteh - a6
* oO
Po Cn ae CAD
AB +CD= RC+AD
Totease cto m of ah
mal intersect at qn’,
1S centre th role and not tho
isorslnishan,
2roRt Leod= =lhod+lh0C = 180"
part [pars pet pons por
Rectangles
DR] pees {neers fetes gan
Pa*+ PD:
(PX omms
=]
Ba
Gn a 0, \lelo AL
2(L 4B) = azsraz
3m @ Rhombus Aacd
dye adie voles
Bit Aas Art Ay
Nek, When the diagonal ineinse:
&
cas:
gt
cD.
TS
aq ahere erp af0
$|
aus (& Pr KB) = avn (AAD)
= bore)
dS
Ls Se ae
aA oh
Cc = (g-2)o o (a-\a-
Fads sf Smaller
Civcde
\e Vs three equal circle,
LS AC_) eSicnas 4
nica smaller orcle
length Rubber bend,
Suceounding equal circle:
length = 34 + LN
d= diamaer:
> + ave (*wap: ABCD) MK
© 4% Paralleleyram,
9} as [qm ABCD =3,
thenor AAPO= 3
e Area distei bution wah’ =>
Area f oye Quod laferss
= J(s-a) (s-»)(s-9(s-4)
Here, 92 gtbtctd .
2
Noted of thora is another circle
but, inardo Quadea bere
Pree of Quod ale ternal
Ge j oonb KiexAe > P 4
© Fh diagonal of a Cyclic qpredailaferal intersect at 90!
i. Ag + apy cp 4 ppt= gt
ie NB a. OC CN EDA eee
i, 4D eRe
° Fre any geyclic qusdaibtaat
* PRXSB= RSX BH4+(ERx PH a
© devon of bile = A Ade i
. longe L diagonal , Wisecls the shex ter ond, ass
i. qo”.
© Polyqont (sequlad)
© Aeeq of henagon = eka
Arona & octagon = o( Km +i)o
Sum of all taterie andes 4 polygen= (n-2) 2180+
«Sumy of all entender angler g Polym= ome)
Total nor ok diagonal fo single, Vener = (n-)
otal: hrolsapalalitigaaals, =j.q 0.052)
2
© Sum of Sea Aste = (Ne 4) ¥ [80 cs ic |
where ne no dh elder bering Bem
Lye oe
Sus =
a Pon 8, Sem b | (C:G)* — aa
L leagth H teanvoue Comme taogeat = leva ay
ae S
De pes 2IRi Rr
SO
fae
fae Ib |
AC o-thon. i |
o
|
Se a a)
Mensuration -3D |
CveE: |
“Total Sunface Areas & a
Uniewalageasen ate
Diagonal = Ba
= 0
———
Volume of Cube
Se ROSults
© when a cube OF moximum vol” $9 Cub peor heralsphore
of Ba sius ‘Wy’, Band the S1de of xb
1 a
. 7 of He cusphere |
vol 9} cube to VO!
fe Two gpheres oh ogual sadtus o*@ daken out
asolid, coke ob sided
Hl cutting +o0m
Find mori um volume jo each sphone:
ki thoy dogonal= gx + 25 ECS be
ard Coece> GA >5ey ; at
Ss ae
ee { Tota! surhece qaoas 2(Lbteh Hh)
Ti Laterof S Avea = a(itb)h
fh Volume =. L%b* 4
Ww Diagond= [Een
© 94 area adjacent haces oftuberd ane ON Ze
1h volume = Saye
ti. Diagonal = (eae
ste 2 ee eel
ii. Volume = (otay-29) (yr wa) (24 -y?)
Cvueold §
2
Cylinder
© Curved surface aren= axth: r
@ Tolal Surpace aren = anr(h+%)
© Volume & cy Under esi
s Diagonal = oa “ae
= juneqh
© Diagona) of Section cut = Fez aRe La
) owt then a veatande \s vlad, it
ay breadth, Ja cyUnder berry
anc= b \* simelarily when it ts
Were Whled along length:
Coze
SN
Radiw= b Ya divs=L *
height = ne Koght = b. » volume of hollow cyftndon = eee
pa a ARThH+ WHO + ox (2-0)
o when Cond Ie Cut sud boom eatin, then
otal Nr g}oed 425 of Cydindes « = onsht i ane
Cone
e volume d Cone = {71h Js
3 ]
© eS h Shlone = woh Ly
e Te Ss Av ok Cone = wilt mt
= Wr (149)
Frus 3 |
e@ Nolume ef frustums Lmh(e 4 vet Re) ve |
@ latwad Curved Ss Aas 7(st+ OL 4
OY (ot IRATE -
ae
@ total Surtace aroac 71 (349 ART TR ane ene-
e@ = Molume of Kusem = Ae [her oo Wa \.
Terxahedron.
e Werte eat Qc slantédge
Ti
eo vue GP ot ih,
6 Sz 3 pare
e Cs As @ Sos
4
L
@ Total sucipce axa = Ba
dqe.
@ Slant hesght= Ba, where a is Blant edge
p) 29: cathe, Cubeld, Cylinder et
Prism (Same base, gare *P)
pariiekare Weight th Prism:
Curved surhece ar2q = Base
Volume > Bane avear hel git ae
“WS: Ac Cro A eee Pyvamid.
Volume = | bate aoa x height:
3
=
Mig cellane ous}
Peerer er sue ul a is alwtide on Bt, Dis m
aide BC, then satio of AD? Od Is -
; &
AO. ee 4
OD Cos Cove La tt
® D> t
fk pwoduct of atnanglds Aliitudes wwe} &
PIs
& Dy altitudes of a A ‘ papas
& \ tw Qq AKO ADINS,
Peri meter oh tiangle % Oe ae
BEM Ly 15 3 Tore) wrk fa a = 60:
ioney Atl AT i a
Nee ee ees: aie ae Bile
Wy Filer 4 y ay 4
Wale 20a toe AS a 235
, Te pig ¢
Aron Honge= (Cotal wont
TE pis X Ege XE ge ¥ Eee
= _60%60 * 60X60 ~ | 50 ent
i 2X2 4XE ay a
a
olen (Se Fytles
sian ne Ne
ale VC Petes ee
te tg 1c 2 0 Ss 4
Mes WH Sigh AO:
3, Vie eS ei 9. ae
5 ioe SDP»
st Perimoter= Bo oma Ph | # Covor's Aheorem
g
AE 4 Cb OS aay
C by DC - ee e >
H# Rouths khooserny A E B
as. (Yr
a SDD Sg a
av (2%) Cay eyet) CUZ4 ZH) Czas 4h)
oe Laddet's Theorem
> Stewart's Theorem Theorem
gn 4 48, AB= AC and D isa
point on BC: Th BD 2SempAGB=\26M>
and AD = 8 OM, shen Fhe length 0,
eps? yer 4 ze
@ Distance bP™ oochaa ia
© POLY)
Ws Abcissa
Yo Ordindle ;
an, ee aCR COS OD”
® Polar Co-wrdi nade tprms chopeints = ¢
tan = a
ve Seay 7D ee
@ Distance hormula > SG-40 a) Oe We
— Ls) » (me tnt)
@ Section fpemulah Inernal diviasen = es
Ply) -
* Gyeanal ba Se amet ney ny, SH ny
L
Pm y) man Cerne?
© mia point aviern 5 (wae) > > (et =)
@ Slopoe Cosle col we Warts) 4 k
M22 a
for parallel Une,
fer perpendicular Una, mea =
tend = Most Se ) Also,
at @] slopez
Ae Yoke
ona 7-H
@ Slope form ior Me, mat C
S Token cept Cutan, yp arts
m= Slope, C
@ Type of Lines. Ineontee ob thargle
A
Orme MBE aatbary, (
Gant be += i ours buy es
BETS BO COD
4. Parallel Gnes Ceiesdien (=m 2
at obi 4 2 Inteas
wb Pe ee
a Coineidence lines ie Sle) a
» Ge se Gels 5
Pees ace é
fo Perpendicular Une @ Antercept berm of OG
NOs
| [Oia + Biba =o
ee Ee a
@ distance ba Une antbytce=e,
dz | seta + by +e 4 1 ae
my
ee (ap
4, Unet= ta Fay ti3=0, fom (4,3) i
16+ % =a = $8_4.6 Any.
az | tea 40 am Hoyt C=0
Sara 5
© distance ha line from
orgs (0,0
(aatby+o vara ‘| eal
>
2g Vines on FUy +30 =2, from odin. ston seen
gn). 90. Dla) 20% 6) 35% a(t, vim)
rea 4 yr *3s)
@ Distance behow~en two pa alle) Linos
a= a <4. Une af0 2 84 F1®
Oe k e ox tes |
erie nege ||. 25 = %
a eqs de
XY
=p and
e Je between two Une:
pope MNO = jan — ton
6-64 14 tang: ton%
© Padxs of Stoadght Usea, _ poasing thug osig’ i a os
Spo A= ae, ort &) multly ak a Staeda
as 4 re oy stengsr ne FA ae
an + b' aay =o
