INDICE LAWS SURD LAWS AREA AND VOLUME
COMMON INDICES SURDS WITH COMMON “BASES” Perimeter Area Volume
a0 = 1 (where a ≠ 0 [= indeterminate] ) b √𝑎 + c √𝑎 = (b + c) √𝑎 CSA - Cross-section Area ℓ - Length
P - Cross-section Perimeter h - Height
a1 = a (where a ≠ 0 [= 0] )
b √𝑎 — c √𝑎 = (b — c) √𝑎 COMMON 2D SHAPES
1 = a— n (where a ≠ 0 [= 0])
Triangle a+b+c xb xh
an
Quadrilateral 2 ( b + h) bxh
b √𝑎 x c √𝑎 = (b x c) √𝑎 = (b . c) √𝑎
(when the WHOLE is a reciprocal = BASE TO −𝑣𝑒 INDICE) Trapezium a+b+c+d (a + c) x h
Circle 2𝜋𝑟 πr2
1
Circle Sector
2𝜋𝑟 x 𝜗 πr2 x 𝜗
an = n
√𝑎 (where a ≥ 0) b √𝑎 ÷ c √𝑎 = ( ). √𝑎 360 360
COMMON 3D SHAPES
(when the INDICE is a reciprocal = ROOT OF BASE) Cube 2 (ab x ac x bc) axbxc
( n√𝑎) n = n
(𝑎) n = a1 = a Cube prism 2 x CSA + P.ℓ (a x b) ℓ
2 (a x b) + P.ℓ (a x b) ℓ
ADD AND SUBTRACT “LIKE BASE AND POWER” INDICES m
( n√𝑎) m = n
(𝑎)m = am/n = a n Triangle prism 2 x CSA + P.ℓ ( b x h) ℓ
𝑥.am + 𝑦.am = (𝑥 + 𝑦) am (like indices)
2 ( b x h) + P.ℓ ( b x h) ℓ
𝑥.am — 𝑦.am = (𝑥 — 𝑦) am (like indices) b x h + P.ℓ
SURDS WITH DIFFERENT “BASES” Cylinder prism 2 x CSA + P.ℓ (𝜋 r2 )ℓ
2 (𝜋r2) + (2𝜋𝑟)ℓ (𝜋 r2 )ℓ
MULTIPLY AND DIVIDE “LIKE BASE” INDICES √𝑎 x √𝑏 = √𝑏 𝑥 𝑎 * Area = 2 x CSA + P . ℓ Volume = CSA x ℓ (for ALL prisms)
m n m n m+n
a .a = a xa = a
√𝑎 ÷ √𝑏 = Pyramid SA of each side base A x h
a m
= am
÷a n
= a m—n Cone SA of each side base A x h
an 2
SIMPLIFYING SURDS FRACTIONS (𝜋rℓ) + (𝜋r ) (𝜋r2) h
a —m
= a —m
x 1 = —m
a .a—n
= a — n + ( — m)
= a —m —n = 𝜋 r (ℓ + r)
an an b . √𝑎 multiple the bottom by an equal Sphere SA of side base A x r
√𝑎 √𝑎 surd to eliminate with a square
2
4(𝜋r ) (𝜋r2) r
RAISING A POWER OF “LIKE BASE” INDICES = b√𝑎
a
4𝜋r2 (𝜋r3)
Hemisphere SA round + top Vol of round
(a m) n = (a n) m = a mxn
4𝜋r2 + 𝜋r2 x (𝜋r3)
a . (√𝑏 − √𝑐) use difference of 2 squares to
(√𝑏 + √𝑐) (√𝑏 − √𝑐) eliminate the surd “root” 3𝜋r2 (𝜋r3)
RECIPROCALS OF “LIKE BASE” INDICES
- always use opposite sign
PROPORTION - PERIMETER / AREA / VOLUME
m 1 1 m m
a n
= a n
= a n
= a . (√𝑏 − √𝑐) simplfy
√𝑏2 + √𝑏 √𝑐 − √𝑏 √𝑐 − √𝑐2 Perimeter A = a1 = b1 Area A = (a1)2 = (b1)2
Perimeter A1 = a2 = b2 Area A1 = (a2)2 = (b2)2
m x 1 m
= n
(𝑎 m) = ( n√𝑎) m = a n
= an a . ( √𝑏 − √𝑐 ) eliminate “like” squares and
* a and b being lengths / sides of a shape
√𝑏2 − √𝑐2 roots
Vol 1 = (Perimeter 1) 3 = Area 1) 2
a . ( √𝑏 − √𝑐 ) Vol 2 = (Perimeter 2) 3 = (Area 2) 2
𝑏 −𝑐
cm x cm x cm = [ cm3 ] cm x cm = [ cm2 ] cm = [ cm ]
, ANGLE LAWS ANGLE LAWS GENERAL
DEFINITIONS POLYGON ANGLES n = # of sides STANDARD FORM
Acute angle 0° < 𝜗 < 90° a x 10n (where 1≤ a < 10;
n = integer)
Right angle 90° = 𝜗
PERCENTAGE CHANGES
Obtuse angle 90° < 𝜗 < 180°
Original +/- change = New
Straight line 180° = 𝜗
Sum of Interior angles I = (n – 2) x 180°
Reflex angle 180° < 𝜗 < 360° Increase value = 𝜗 + Δ % . 𝜗
I + E = 180° x n 100
[(n – 2) x 180°] + E = 180° x n
ANGLE RULES E = 360°
Decrease value = 𝜗 – Δ % . 𝜗
Complementary angle sum adds up to 90° ∴ 100
Sum of Supplementary Exterior angles E = 360°
Supplementary angle sum adds up to 180° (always true)
BEARINGS
Angles around a point sum adds up to 360° TRIANGLE ANGLES → (n = 3)
Always calculate bearing from the NORTH and CLOCKWISE.
Triangle – sum of interior add up to 180°
If a bearing is X to Y , then to calculate Y to X
Triangle – sum of full exterior add up to 900° Then add 180° (but if answer is greater than 360° then minus 180°
Triangle exterior sum of two opposite angles
Vertically opposite angles are equal X a+b=c North
Corresponding angles are equal F
b North
Alternate angles are equal Z X
a c
Co-Interior angles add up to 180° C
Y
QUADRILATERAL ANGLES → (n = 4)
Quadrilateral – sum of interior add up to 360° CONGRUENT
Quadrilateral – sum of full exterior add up to 1080° Same shape
Same size sides
Same size angles
Note Supplementary exterior ≠ Full exterior
COMMON INDICES SURDS WITH COMMON “BASES” Perimeter Area Volume
a0 = 1 (where a ≠ 0 [= indeterminate] ) b √𝑎 + c √𝑎 = (b + c) √𝑎 CSA - Cross-section Area ℓ - Length
P - Cross-section Perimeter h - Height
a1 = a (where a ≠ 0 [= 0] )
b √𝑎 — c √𝑎 = (b — c) √𝑎 COMMON 2D SHAPES
1 = a— n (where a ≠ 0 [= 0])
Triangle a+b+c xb xh
an
Quadrilateral 2 ( b + h) bxh
b √𝑎 x c √𝑎 = (b x c) √𝑎 = (b . c) √𝑎
(when the WHOLE is a reciprocal = BASE TO −𝑣𝑒 INDICE) Trapezium a+b+c+d (a + c) x h
Circle 2𝜋𝑟 πr2
1
Circle Sector
2𝜋𝑟 x 𝜗 πr2 x 𝜗
an = n
√𝑎 (where a ≥ 0) b √𝑎 ÷ c √𝑎 = ( ). √𝑎 360 360
COMMON 3D SHAPES
(when the INDICE is a reciprocal = ROOT OF BASE) Cube 2 (ab x ac x bc) axbxc
( n√𝑎) n = n
(𝑎) n = a1 = a Cube prism 2 x CSA + P.ℓ (a x b) ℓ
2 (a x b) + P.ℓ (a x b) ℓ
ADD AND SUBTRACT “LIKE BASE AND POWER” INDICES m
( n√𝑎) m = n
(𝑎)m = am/n = a n Triangle prism 2 x CSA + P.ℓ ( b x h) ℓ
𝑥.am + 𝑦.am = (𝑥 + 𝑦) am (like indices)
2 ( b x h) + P.ℓ ( b x h) ℓ
𝑥.am — 𝑦.am = (𝑥 — 𝑦) am (like indices) b x h + P.ℓ
SURDS WITH DIFFERENT “BASES” Cylinder prism 2 x CSA + P.ℓ (𝜋 r2 )ℓ
2 (𝜋r2) + (2𝜋𝑟)ℓ (𝜋 r2 )ℓ
MULTIPLY AND DIVIDE “LIKE BASE” INDICES √𝑎 x √𝑏 = √𝑏 𝑥 𝑎 * Area = 2 x CSA + P . ℓ Volume = CSA x ℓ (for ALL prisms)
m n m n m+n
a .a = a xa = a
√𝑎 ÷ √𝑏 = Pyramid SA of each side base A x h
a m
= am
÷a n
= a m—n Cone SA of each side base A x h
an 2
SIMPLIFYING SURDS FRACTIONS (𝜋rℓ) + (𝜋r ) (𝜋r2) h
a —m
= a —m
x 1 = —m
a .a—n
= a — n + ( — m)
= a —m —n = 𝜋 r (ℓ + r)
an an b . √𝑎 multiple the bottom by an equal Sphere SA of side base A x r
√𝑎 √𝑎 surd to eliminate with a square
2
4(𝜋r ) (𝜋r2) r
RAISING A POWER OF “LIKE BASE” INDICES = b√𝑎
a
4𝜋r2 (𝜋r3)
Hemisphere SA round + top Vol of round
(a m) n = (a n) m = a mxn
4𝜋r2 + 𝜋r2 x (𝜋r3)
a . (√𝑏 − √𝑐) use difference of 2 squares to
(√𝑏 + √𝑐) (√𝑏 − √𝑐) eliminate the surd “root” 3𝜋r2 (𝜋r3)
RECIPROCALS OF “LIKE BASE” INDICES
- always use opposite sign
PROPORTION - PERIMETER / AREA / VOLUME
m 1 1 m m
a n
= a n
= a n
= a . (√𝑏 − √𝑐) simplfy
√𝑏2 + √𝑏 √𝑐 − √𝑏 √𝑐 − √𝑐2 Perimeter A = a1 = b1 Area A = (a1)2 = (b1)2
Perimeter A1 = a2 = b2 Area A1 = (a2)2 = (b2)2
m x 1 m
= n
(𝑎 m) = ( n√𝑎) m = a n
= an a . ( √𝑏 − √𝑐 ) eliminate “like” squares and
* a and b being lengths / sides of a shape
√𝑏2 − √𝑐2 roots
Vol 1 = (Perimeter 1) 3 = Area 1) 2
a . ( √𝑏 − √𝑐 ) Vol 2 = (Perimeter 2) 3 = (Area 2) 2
𝑏 −𝑐
cm x cm x cm = [ cm3 ] cm x cm = [ cm2 ] cm = [ cm ]
, ANGLE LAWS ANGLE LAWS GENERAL
DEFINITIONS POLYGON ANGLES n = # of sides STANDARD FORM
Acute angle 0° < 𝜗 < 90° a x 10n (where 1≤ a < 10;
n = integer)
Right angle 90° = 𝜗
PERCENTAGE CHANGES
Obtuse angle 90° < 𝜗 < 180°
Original +/- change = New
Straight line 180° = 𝜗
Sum of Interior angles I = (n – 2) x 180°
Reflex angle 180° < 𝜗 < 360° Increase value = 𝜗 + Δ % . 𝜗
I + E = 180° x n 100
[(n – 2) x 180°] + E = 180° x n
ANGLE RULES E = 360°
Decrease value = 𝜗 – Δ % . 𝜗
Complementary angle sum adds up to 90° ∴ 100
Sum of Supplementary Exterior angles E = 360°
Supplementary angle sum adds up to 180° (always true)
BEARINGS
Angles around a point sum adds up to 360° TRIANGLE ANGLES → (n = 3)
Always calculate bearing from the NORTH and CLOCKWISE.
Triangle – sum of interior add up to 180°
If a bearing is X to Y , then to calculate Y to X
Triangle – sum of full exterior add up to 900° Then add 180° (but if answer is greater than 360° then minus 180°
Triangle exterior sum of two opposite angles
Vertically opposite angles are equal X a+b=c North
Corresponding angles are equal F
b North
Alternate angles are equal Z X
a c
Co-Interior angles add up to 180° C
Y
QUADRILATERAL ANGLES → (n = 4)
Quadrilateral – sum of interior add up to 360° CONGRUENT
Quadrilateral – sum of full exterior add up to 1080° Same shape
Same size sides
Same size angles
Note Supplementary exterior ≠ Full exterior
