Steply

Geometry formula sheet

62 formulas · what each one means · a free lesson for each

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Angles

Angles on a line
a + b = 180°
Angles on a straight line add up to 180°.Lesson: Angle facts →
Angles around a point
a + b + c + … = 360°
A full turn is 360°.Lesson: Angle facts →
Angles in a triangle
A + B + C = 180°
Lesson: Angle facts →
Vertical angles
opposite angles are equal
Two crossing lines make equal opposite angles.Lesson: Angle facts →
Complementary and supplementary
a + b = 90° a + b = 180°
Complementary add to 90°; supplementary add to 180°.Lesson: Complementary and supplementary angles →
Exterior angle of a triangle
d = A + B
The outside angle equals the two far inside angles added.Lesson: Exterior angle of a triangle →
Triangle inequality
a + b > c
The two short sides together must be longer than the longest side.Lesson: Can it make a triangle →
Parallel lines
corresponding = · alternate = · co-interior add to 180°
F shape: equal. Z shape: equal. C shape: add to 180°.Lesson: Parallel lines and a transversal →
Angle sum of a polygon
(n − 2) × 180°
n is the number of sides.Lesson: Angles in a polygon →
One angle of a regular polygon
(n − 2) × 180°n
All angles are equal, so share the sum.Lesson: Angles in a polygon →
Exterior angles
sum = 360° one (regular) = 360°n
Walking around a polygon turns you once.Lesson: Exterior angles →

Perimeter and area

Rectangle
P = 2l + 2w A = lw
Perimeter = all the way around; area = length × width.Lesson: Perimeter and area →
Square
P = 4s A = s2
Lesson: Perimeter and area →
Triangle
A = 12bh
Half of base × height; the height meets the base at 90°.Lesson: Perimeter and area →
Parallelogram
A = bh
Base × height (the straight-up height, not the slanted side).Lesson: Parallelogram and trapezoid →
Trapezoid
A = 12h(b1 + b2)
The mean of the two parallel sides × the height.Lesson: Parallelogram and trapezoid →
Rhombus and kite
A = 12d1 d2
Half of one diagonal × the other.Lesson: Rhombus and kite →
Circle: circumference
C = 2πr = πd
The distance around a circle.Lesson: Perimeter and area →
Circle: area
A = πr2
Lesson: Perimeter and area →
Arc length
θ360° × 2πr
The arc is a part of the circumference: θ out of 360°.Lesson: Arc length, sector area and perimeter as a part of the circle →
Sector area
θ360° × πr2
The sector is a part of the circle's area.Lesson: Arc length, sector area and perimeter as a part of the circle →
Composite shapes
add the parts, take away the holes
Split the shape into rectangles, triangles and circles.Lesson: Composite shapes →

Volume and surface area

Rectangular box
V = lwh SA = 2lw + 2lh + 2wh
Surface area = the six faces added.Lesson: Volume and surface area →
Cube
V = s3 SA = 6s2
Lesson: Volume and surface area →
Prism
V = Bh
B is the area of the end shape, h the length between the ends.Lesson: Prisms →
Cylinder
V = πr2 h SA = 2πr2 + 2πrh
Two circles plus the curved side (a rectangle when unrolled).Lesson: Volume and surface area →
Pyramid
V = 13Bh
One third of the prism with the same base and height.Lesson: Pyramids →
Cone
V = 13πr2 h SA = πr2 + πrl
l is the slant height (side, from tip to edge).Lesson: Cones →
Sphere
V = 43Ï€r3 SA = 4Ï€r2
Lesson: Spheres and hemispheres →
Hemisphere
V = 23Ï€r3 SA = 3Ï€r2
Half a sphere plus the flat circle (2πr² + πr²).Lesson: Spheres and hemispheres →
Frustum
V = big cone − small cone
A cone with its top cut off.Lesson: Volume of a frustum →
Similar solids
lengths × k, areas × k2, volumes × k3
k is the scale factor.Lesson: Similar shapes and solids →

Right triangles

Pythagorean theorem
a2 + b2 = c2
c is the hypotenuse, the long side across from the right angle.Lesson: Pythagoras →
Is it a right triangle?
a2 + b2 = c2 ⇔ right angle
If the two short sides squared add to the long side squared.Lesson: Is it a right triangle? →
45-45-90 triangle
x, x, x√2
Legs equal; hypotenuse = leg × √2.Lesson: Special right triangles →
30-60-90 triangle
x, x√3, 2x
Short leg x (across 30°), long leg x√3, hypotenuse 2x.Lesson: Special right triangles →
Sine
sin θ = oppositehypotenuse
SOHLesson: Right-triangle trigonometry →
Cosine
cos θ = adjacenthypotenuse
CAHLesson: Right-triangle trigonometry →
Tangent
tan θ = oppositeadjacent
TOALesson: Right-triangle trigonometry →
Cofunctions
sin x = cos(90° − x)
The sine of one acute angle is the cosine of the other.Lesson: sin of one angle = cos of the other →
Diagonal of a box
d = √l2 + w2 + h2
Pythagoras twice: floor diagonal, then up.Lesson: Pythagoras in 3D →

Coordinate geometry

Distance formula
d = √(x2 − x1)2 + (y2 − y1)2
Pythagoras on the grid.Lesson: Distance between two points →
Midpoint
M = (x1 + x22, y1 + y22)
The mean of the x's and the mean of the y's.Lesson: Midpoint →
Circle
(x − h)2 + (y − k)2 = r2
Center (h, k), radius r.Lesson: Equation of a circle →

Similar and congruent triangles

Similar triangles
ad = be = cf
Same shape: matching sides are in the same ratio.Lesson: Similar triangles →
Scale drawings
real length = drawing length × scale
1 cm : 5 km means each cm is 5 km.Lesson: Scale drawings and maps →
Congruent triangles
SSS, SAS, ASA, AAS, RHS
Any one of these is enough to prove two triangles are the same.Lesson: Congruent triangles →
Isosceles triangle
two equal sides ⇔ two equal base angles
Lesson: Isosceles triangles →

Circle rules

Angle at the center
center = 2 × edge
The angle at the center is twice the angle at the edge on the same arc.Lesson: Circle rules →
Angle in a semicircle
90°
An angle standing on a diameter is a right angle.Lesson: Circle rules →
Same segment
angles on the same arc are equal
Lesson: Circle rules →
Cyclic quadrilateral
A + C = 180° B + D = 180°
Opposite angles add to 180°.Lesson: Cyclic quadrilaterals →
Tangent and radius
90°
A tangent meets the radius at a right angle.Lesson: Tangents →
Two tangents from a point
equal lengths
Lesson: Tangents →
Intersecting chords
a × b = c × d
The two parts of one chord multiply to the same as the other's.Lesson: Intersecting chords, secants and tangents →

Transformations

Translation
(x, y) → (x + a, y + b)
Slide every point the same way.Lesson: Translations →
Reflect in the x-axis
(x, y) → (x, −y)
Lesson: Reflections →
Reflect in the y-axis
(x, y) → (−x, y)
Lesson: Reflections →
Reflect in y = x
(x, y) → (y, x)
Swap the coordinates.Lesson: Reflections →
Rotate 90° counterclockwise
(x, y) → (−y, x)
About the origin. 90° clockwise: (y, −x).Lesson: Rotations about the origin →
Rotate 180°
(x, y) → (−x, −y)
About the origin.Lesson: Rotations about the origin →
Dilation
(x, y) → (kx, ky)
From the origin, scale factor k.Lesson: Dilations about the origin →

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