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Midpoint Formula
Given points a and b on a number line.
M(2a+b)
M(2a−b)
M(a,b)
M(2b−a)
Midpoint Formula
On a plane with endpoints (x1, y1) and (x2, y2) .
d=((x2− x1)2+(y2− y1)2)
M(x2− x1, y2− y1)
M(2x1+x2, 2y1+y2)
M(2x1−x2, 2y1−y2)
Distance Formula
On a plane with endpoints (x2, y2) and (x2, y2) .
d=(x1−y1)2+(x2−y2)2
d=(x2−y1)2+(x1−y2)2
d=(x2+x1)2−(y2+y1)2
d=(x2−x1)2+(y2−y1)2
Perimeter of a Square
P=4s
P=s2
P=a+b+c
P=s
Area of a Square
A=4s
A=s2
A=21bh
A=21h(b1+b2)
Perimeter of a Rectangle
P=4s
P=s2
P=2b+2h
P=bh
Area of a Rectangle
A=4s
A=s2
A=2b+2h
A=bh
Perimeter of a Triangle
P=4s
P=a+b+c
P=21bh
P=s
Area of a Triangle
A=4s
A=bh
A=21bh
A=21h(b1+b2)
Circumference of a Circle
C=dπ
C=πr2
C=2πr
C=2πr2
Area of a Circle
A=dπ
A=πr2
A=2πr
A=2πr2
Converse of a→b
b→a
∼a→∼b
∼b→∼a
a→b
Inverse of a→b
b→a
∼a→∼b
∼b→∼a
a→b
Contrapositive of a→b
∼b→∼a
∼a→∼b
b→a
a→b
Triangle Angle-Sum Theorem
m∠A+m∠B+m∠C = 360°
n(n−2)180
a2+b2=c2
m∠A+m∠B+m∠C = 180°
Triangle Exterior Angle Theorem
m∠1=180−(m∠2+m∠3)
m∠1=m∠2−m∠3
m∠1=m∠2+m∠3
m∠1=180−m∠4
Reflexive Property of Congruence
∠A≅∠A
If ∠A≅∠B , then ∠A≅∠B
If ∠A≅∠B and ∠B≅∠C , then ∠A≅∠C
If ∠A≅∠B , then ∠B≅∠C
Symmetric Property of Congruence
∠A≅∠A
If ∠A≅∠B , then ∠A≅∠B
If ∠A≅∠B and ∠B≅∠C , then ∠A≅∠C
If ∠A≅∠B , then ∠B≅∠C
Transitive Property of Congruence
∠A≅∠A
If ∠A≅∠B , then ∠A≅∠B
If ∠A≅∠B and ∠B≅∠C , then ∠A≅∠C
If ∠A≅∠B , then ∠B≅∠C
Slope Formula
On a plane with endpoints (x1, y1) and (x2, y2) .
m=x2−x1y2−y1
m=y2−y1x2−x1
M(2x1+x2, 2y1+y2)
M(2x1−x2, 2y1−y2)
Polygon Angle-Sum Theorem
(n−2)180
2(n−2)180
n(n−2)180
n−2⋅180
Corollary to the Polygon Angle-Sum Theorem
(n−2)180
2(n−2)180
n(n−2)180
n−2⋅180
Slope Intercept Form
y−y1=m(x−x1)
(x−h)2+(y−k)2=r2
y=mx+b
Ax+By=C
Point-Slope Form
y−y1=m(x−x1)
(x−h)2+(y−k)2=r2
y=mx+b
Ax+By=C
Polygon Exterior Angle-Sum Theorem
The sum of the interior angles of a polygon is 180°
The sum of the interior angles of a polygon is 360°
The sum of the exterior angles of a polygon is 180°
The sum of the exterior angles of a polygon is 360°
Trapezoid Midsegment Theorem
The midsegment of a trapezoid is the sum of the bases.
The midsegment of a trapezoid is the average of the bases.
The midsegment of a trapezoid is twice the size of the smaller base.
The midsegment of a trapezoid is half the size of the larger base.
Pythagorean Theorem
a2−b2=c2
Hypotenuse=2⋅Short Leg
Hypotenuse=Leg⋅2
a2+b2=c2
45°−45°−90° Triangle Theorem
Hypotenuse=2⋅Short Leg
Hypotenuse=Leg⋅2
Hypotenuse=Short Leg⋅3
Long Leg=Short Leg⋅3
30°−60°−90° Triangle Theorem
Hypotenuse=2⋅Short Leg
Hypotenuse=Leg⋅2
Hypotenuse=Short Leg⋅3
Long Leg=Short Leg⋅3
Sine ratio for right triangles
sin(angle)=adjacent legopposite leg
sin(angle)=hypotenuseadjacent leg
sin(angle)=hypotenuseopposite leg
sin(angle)=opposite legadjacent leg
Cosine ratio for right triangles
cos(angle)=adjacent legopposite leg
cos(angle)=hypotenuseadjacent leg
cos(angle)=hypotenuseopposite leg
cos(angle)=opposite legadjacent leg
Tangent ratio for right triangles
tan(angle)=adjacent legopposite leg
tan(angle)=hypotenuseadjacent leg
tan(angle)=hypotenuseopposite leg
tan(angle)=opposite legadjacent leg
90° rotation about the origin.
r(90°,O)(x,y)=(x,y)
r(90°,O)(x,y)=(−x,−y)
r(90°,O)(x,y)=(y,x)
r(90°,O)(x,y)=(−y,x)
r(90°,O)(x,y)=(y,−x)
180° rotation about the origin.
r(180°,O)(x,y)=(x,y)
r(180°,O)(x,y)=(−x,−y)
r(180°,O)(x,y)=(y,x)
r(180°,O)(x,y)=(−y,x)
r(180°,O)(x,y)=(y,−x)
270° rotation about the origin.
r(270°,O)(x,y)=(x,y)
r(270°,O)(x,y)=(−x,−y)
r(270°,O)(x,y)=(y,x)
r(270°,O)(x,y)=(−y,x)
r(270°,O)(x,y)=(y,−x)
360° rotation about the origin.
r(360°,O)(x,y)=(x,y)
r(360°,O)(x,y)=(−x,−y)
r(360°,O)(x,y)=(y,x)
r(360°,O)(x,y)=(−y,x)
r(360°,O)(x,y)=(y,−x)
Area of a Parallelogram
A=4s
A=s2
A=2b+2h
A=bh
Area of a Trapezoid
A=21bh
A=21(b1+b2)
A=21h(b1+b2)
A=21ap
Area of a Rhombus
A=21bh
A=21d1d2
A=21h(b1+b2)
A=21ap
Area of a Kite
A=21bh
A=21d1d2
A=21h(b1+b2)
A=21ap
Area of a Regular Polygon
A=21bh
A=21d1d2
A=21h(b1+b2)
A=21ap
Area of a Triangle (given SAS)
A=21bh
A=21s1s2⋅sin(included angle)
A=2 1bhw
A=21b⋅sin(included angle)
Arc Length
length of arc =360arc measure⋅dπ
length of arc =360arc measure⋅πr2
length of arc = sector area − segment area
length of arc =360πr2
Area of a Sector
Sector Area =360arc measure⋅dπ
Sector Area =360arc measure⋅πr2
Sector Area = sector area − segment area
Sector Area =360πr2
Area of a Segment
Segment Area =360arc measure⋅dπ
Segment Area =360arc measure⋅πr2
Segment Area = sector area − triangle area
Segment Area =360πr2
Euler's Formula
F+E=V+2
F+2=V+E
E+V=F+2
F+V=E+2
Density
D=Vm
D=mV
D=Varea
D=aream
Lateral Area of a Cylinder
LA=πrh
LA=πrl
LA=πdh
LA=πr2h
Surface Area of a Cylinder
SA=πdh
SA=πdh+2πr2
SA=πrl+πr2
SA=πr2h
Volume of a Cylinder
V=πdh
V=πdh+2πr2
V=πr2h
V=34πr3
Lateral Area of a Pyramid
LA=21rh
LA=21pl
LA=21pl+B
LA=pl
Surface Area of a Pyramid
SA=31Bh
SA=21pl
SA=21pl+B
SA=pl
Volume of a Pyramid
V=31Bh
V=21pl
V=21pl+B
V=pl
Lateral Area of a Cone
LA=πrh
LA=πrl
LA=πdh
LA=πr2h
Surface Area of a Cone
SA=πdh+2πr2
SA=πrl
SA=πrl+πr2
SA=31πr2h
Volume of a Cone
V=πdh+2πr2
V=πrl
V=πrl+πr2
V=31πr2h
Surface Area of a Sphere
SA=4πr2
SA=4πr2h
SA=34πr3
SA=31πr2h
Volume of a Sphere
V=4πr2
V=4πr2h
V=34πr3
V=31πr2h
Equation of a circle
with center at (h,k)
(x+h)2−(y+k)2=r2
(x−h)2+(y−k)2=r2
(x−k)2+(y−h)2=r2
(x−h)2−(y−k)2=r2
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