Geometry Basic Flashcards
Sum of angles on a straight line
180°
Sum of angles around a point
360°
Definition of a right angle
90°
Complementary angles sum to
90°
Supplementary angles sum to
180°
Two lines that never intersect (in Euclidean geometry)
Parallel lines
Angles across from each other when two lines intersect
Vertical angles
Sum of vertical angles
They are equal (same measure)
Alternate interior angles when lines are parallel are
Equal
Definition of a transversal line
A line that intersects two or more other lines
Sum of interior angles in a triangle
180°
An isosceles triangle has how many equal sides?
Two
An equilateral triangle has angles measuring
60° each
A right triangle has one angle measuring
90°
Pythagorean theorem formula
a² + b² = c²
Type of triangle with all angles < 90°
Acute triangle
Type of triangle with one angle > 90°
Obtuse triangle
Median of a triangle connects a vertex to
The midpoint of the opposite side
Altitude (height) of a triangle is drawn
Perpendicular to the opposite side
Sum of two sides of a triangle is always
Greater than the third side
Sum of interior angles in a quadrilateral
360°
Rectangle definition
A quadrilateral with four right angles
Square definition
A rectangle with all sides equal
Parallelogram definition
Opposite sides parallel and equal
Rhombus definition
A parallelogram with four equal sides
A trapezoid (US) or trapezium (UK) has
At least one pair of parallel sides
Kite definition (in geometry)
A quadrilateral with two pairs of adjacent equal sides
Opposite angles in a parallelogram are
Equal
Diagonals of a rectangle are
Equal in length
Diagonals of a parallelogram
Bisect each other
A pentagon has how many sides?
5
A hexagon has how many sides?
6
A heptagon (or septagon) has how many sides?
7
Sum of interior angles of a pentagon
540°
Sum of interior angles of a hexagon
720°
Formula for sum of interior angles of an n-sided polygon
(n - 2) × 180°
A regular polygon has
All sides and angles equal
Number of diagonals in an n-sided polygon
n(n - 3)/2
Interior angle of a regular quadrilateral (square)
90°
Interior angle of a regular triangle (equilateral)
60°
Circumference formula
C = 2πr
Area of a circle formula
A = πr²
Diameter of a circle
2r
Definition of a chord
Line segment joining two points on a circle
Definition of a secant
Line that intersects a circle in two points
Definition of a tangent to a circle
Line that touches the circle at exactly one point
Tangents from a common external point to a circle are
Equal in length
Definition of a central angle
An angle whose vertex is the center of the circle
Definition of an inscribed angle
An angle formed by two chords with the vertex on the circle
Inscribed angle theorem
An inscribed angle is half the measure of its intercepted arc
Slope formula between (x₁, y₁) and (x₂, y₂)
(y₂ - y₁) / (x₂ - x₁)
Midpoint formula
((x₁ + x₂)/2, (y₁ + y₂)/2)
Distance formula
√[(x₂ - x₁)² + (y₂ - y₁)²]
Equation of a line in slope-intercept form
y = mx + b
Standard form of a line
Ax + By = C
Slope of x-axis
0
Slope of y-axis
Undefined (vertical line)
Point-slope form of a line
y - y₁ = m(x - x₁)
Length of the vector (a, b)
√(a² + b²)
Equation of a circle in center-radius form
(x - h)² + (y - k)² = r²
Translation moves a shape
Without rotation; shifts every point by the same distance
Reflection definition
A flip over a line or axis producing a mirror image
Rotation definition
Turning a figure about a fixed point (the center of rotation)
90° rotation about the origin (x, y)
(-y, x)
180° rotation about the origin (x, y)
(-x, -y)
Dilation definition
Enlarges or reduces a figure using a scale factor
Scale factor = 2 means
All distances double from the center of dilation
Invariant point under reflection across x-axis
A point on the x-axis (y=0)
Invariant point under reflection across y-axis
A point on the y-axis (x=0)
Composition of transformations
Applying multiple transformations sequentially
Cube definition
A prism with six square faces
Rectangular prism definition
A prism with rectangular faces
Cylinder definition
A solid with two congruent circular bases connected by a curved surface
Sphere definition
A set of points in 3D at a fixed distance (radius) from a center
Cone definition
A solid with a circular base and a vertex (apex) not in the plane of the circle
Pyramid definition
A solid with a polygon base and triangular faces meeting at an apex
Tetrahedron definition
A pyramid with a triangular base (4 triangular faces)
Edges in a cube
12
Faces in a rectangular prism
6
Euler’s formula for polyhedra
V - E + F = 2
Perimeter of a square with side s
4s
Perimeter of a rectangle with length L and width W
2(L + W)
Area of a square with side s
s²
Area of a rectangle with length L and width W
LW
Area of a parallelogram
Base × Height
Area of a triangle
½ (Base × Height)
Surface area of a cube with side s
6s²
Volume of a cube with side s
s³
Volume of a rectangular prism with length L, width W, height H
LWH
Volume of a cylinder with radius r and height h
πr²h
Volume of a cone
(1/3)πr²h
Volume of a sphere
(4/3)πr³
Surface area of a sphere
4πr²
Diagonals of a rectangle
Are equal and bisect each other
Inradius of a right triangle with legs a, b, and hypotenuse c
(a + b - c)/2
Triangle inequality property
Sum of any two sides > third side
Exterior angle of a triangle equals
Sum of the two non-adjacent interior angles
Altitude to the hypotenuse in a right triangle forms
Two similar subtriangles to the original
Midsegment of a triangle (connecting midpoints of two sides)
Parallel to the third side and half its length
Lateral surface area of a cylinder
2πrh
Define a line segment in geometry.
A portion of a line bounded by two distinct endpoints.
Define a ray in geometry.
A part of a line starting at one endpoint and extending infinitely in one direction.
What is a transversal line?
A line that intersects two or more lines at distinct points.
What does it mean for two angles to be adjacent?
They share a common vertex and a common side but do not overlap.
Define a linear pair of angles.
Two adjacent angles whose non-common sides form a straight line (sum is 180°).
What is a convex polygon?
A polygon with all interior angles less than 180°, and vertices ‘point outward.’
What is a concave polygon?
A polygon with at least one interior angle greater than 180°, and a ‘dent’ in its shape.
Define the term ‘diagonal’ in a polygon.
A line segment connecting two non-adjacent vertices.
In geometry, what is a midsegment of a trapezoid?
A segment that joins the midpoints of the non-parallel sides; its length is the average of the parallel sides.
Define an exterior angle of a polygon.
An angle formed by extending one side of the polygon at a vertex.
State the Exterior Angle Theorem (for triangles).
An exterior angle of a triangle equals the sum of the two non-adjacent interior angles.
What is the centroid of a triangle?
The point where the three medians intersect (it divides each median in a 2:1 ratio).
Define the incenter of a triangle.
The point where the three angle bisectors intersect; center of the inscribed circle.
Define the circumcenter of a triangle.
The point where the perpendicular bisectors of the sides intersect; center of the circumscribed circle.
Define the orthocenter of a triangle.
The point where the three altitudes intersect.
State the formula for the perimeter of an equilateral triangle with side s.
P = 3s.
Define the concept of similar triangles.
Triangles that have the same shape but not necessarily the same size (corresponding angles equal, sides in proportion).
State the Triangle Proportionality Theorem.
A line parallel to one side of a triangle divides the other two sides proportionally.
What is the formula for the area of an isosceles triangle with base b and height h?
A = (1/2) × b × h.
Define the circumscribed circle of a triangle.
A circle passing through all three vertices of the triangle.
What does AAA similarity criterion stand for?
Angle-Angle-Angle: if three angles of one triangle match three angles of another, the triangles are similar.
Define the altitude of a trapezoid.
The perpendicular distance between the two parallel sides.
What is a regular polygon?
A polygon with all sides and angles equal (e.g., regular pentagon, regular hexagon).
Define an n-sided polygon’s interior angle measure if it is regular.
Each interior angle = [(n - 2) × 180°] / n.
State the formula for finding the total number of diagonals in an n-sided polygon.
[n(n - 3)] / 2.
Define a central angle in a regular polygon.
An angle formed by two radii drawn to consecutive vertices of the polygon.
What is the measure of each central angle in a regular n-sided polygon?
360° / n.
Define the apothem of a regular polygon.
A line from the center perpendicular to any side, used in finding area.
Write the area formula for a regular polygon with perimeter p and apothem a.
Area = (1/2) × p × a.
Define a frustum of a cone.
The shape obtained by slicing a cone parallel to its base and removing the top portion.
Define an arc of a circle.
A portion of the circumference between two points on the circle.
What is the measure of a minor arc?
Equal to the measure of the central angle that subtends it, in degrees.
What is the formula for arc length of a circle with radius r and subtended angle θ (in degrees)?
Arc length = (θ/360°) × 2πr.
Define a major arc in a circle.
An arc larger than a semicircle, more than 180°.
Define a semicircle in a circle.
An arc that measures exactly 180°.
What is a chord’s perpendicular bisector property in a circle?
It passes through the center of the circle.
Explain the inscribed angle theorem in circles.
The measure of an inscribed angle is half the measure of its intercepted arc.
What is the formula for the area of a sector of a circle with radius r and angle θ (in degrees)?
Area of sector = (θ/360°) × πr².
Define a segment of a circle (minor segment).
A region formed by a chord and the arc it subtends.
Write the area formula for the segment of a circle (minor segment) with chord AB and radius r.
Area of segment = Area of sector - Area of triangle formed by AB and center.
Define a tangent to a circle from a point A outside the circle.
A line passing through A that touches the circle at exactly one point.
What is the tangent-secant theorem?
Tangent² = External segment × Whole secant (if a secant is drawn from the same external point).
What is the power of a point theorem for a circle?
For a point P outside the circle: (PT)² = PA × PB (if T is tangent, A and B are intersection points of a secant).
Define two tangents from an external point to a circle.
They are congruent (equal in length).
What is the measure of angles formed by tangents from an external point?
The tangents form an isosceles triangle with the line connecting the circle’s center and external point.
Define concentric circles.
Circles with the same center but different radii.
Write the distance formula for two points (x₁, y₁) and (x₂, y₂) in the plane.
√[(x₂ - x₁)² + (y₂ - y₁)²].
What is the midpoint formula in the Cartesian plane for points (x₁, y₁) and (x₂, y₂)?
((x₁ + x₂)/2, (y₁ + y₂)/2).
Define the slope of a line in coordinate geometry.
(y₂ - y₁) / (x₂ - x₁), for x₂ ≠ x₁.
Write the equation of a circle with center (h, k) and radius r.
(x - h)² + (y - k)² = r².
Define the equation of a horizontal line through (x₀, y₀).
y = y₀.
Define the equation of a vertical line through (x₀, y₀).
x = x₀.
What is the formula for the sum of the interior angles of an n-sided polygon?
(n - 2) × 180°
State the formula for the measure of each interior angle in a regular n-sided polygon.
[(n - 2) × 180°] / n
What is the perimeter formula for a regular n-sided polygon with side length s?
P = n × s
Give the formula for the sum of exterior angles of any convex polygon.
360°
State the formula for the measure of each exterior angle of a regular n-sided polygon.
360° / n
Formula for the slope m between two points (x₁, y₁) and (x₂, y₂) in the plane?
m = (y₂ - y₁) / (x₂ - x₁)
Write the distance formula between (x₁, y₁) and (x₂, y₂) in 2D coordinate geometry.
d = √[(x₂ - x₁)² + (y₂ - y₁)²]
Give the midpoint formula between (x₁, y₁) and (x₂, y₂).
((x₁ + x₂)/2, (y₁ + y₂)/2)
Formula for the equation of a line in point-slope form?
y - y₁ = m(x - x₁)
Formula for the equation of a line in slope-intercept form?
y = mx + b
Standard form of a line equation in the plane?
Ax + By = C
Circumference formula for a circle of radius r?
C = 2πr
Area formula for a circle of radius r?
A = πr²
Arc length formula for a circle with radius r subtended by central angle θ (in radians)?
Arc length = rθ
Area of a sector formula for angle θ (in radians) in a circle of radius r?
Area of sector = (θ/2)r²
Formula for the length of a chord in a circle with radius r, subtended by central angle θ (in radians)?
Chord length = 2r sin(θ/2)
Area of an ellipse with semi-major axis a and semi-minor axis b?
A = πab
Distance formula from a point (x₀, y₀) to the line Ax + By + C = 0?
Distance = |Ax₀ + By₀ + C| / √(A² + B²)
Formula for the slope of a perpendicular line to y = m₁x + b₁?
m₂ = -1/m₁
Sum of the interior angles of a triangle?
180°
Perimeter formula for a triangle with sides a, b, c?
P = a + b + c
Formula for the area of a triangle given base b and height h?
A = ½ × b × h
Heron’s formula for the area of a triangle with sides a, b, c?
A = √[s(s - a)(s - b)(s - c)], where s = (a + b + c)/2
Formula for area of an equilateral triangle with side s?
A = (√3 / 4) s²
Pythagorean theorem for a right triangle with legs a, b and hypotenuse c?
a² + b² = c²
Perimeter of a rectangle with length ℓ and width w?
P = 2(ℓ + w)
Area of a rectangle with length ℓ and width w?
A = ℓ × w
Formula for the diagonal of a rectangle with length ℓ and width w?
d = √(ℓ² + w²)
Perimeter of a square with side s?
P = 4s
Area of a square with side s?
A = s²
Perimeter of a parallelogram with sides a, b?
P = 2(a + b)
Area of a parallelogram with base b and height h?
A = b × h
Perimeter of a rhombus with side s?
P = 4s
Area of a rhombus using diagonals d₁ and d₂?
A = ½(d₁ × d₂)
Area of a trapezoid (US) or trapezium (UK) with bases b₁, b₂ and height h?
A = ½(b₁ + b₂) × h
Sum of the interior angles of a quadrilateral?
360°
Inradius (r) formula for a right triangle with legs a, b and hypotenuse c?
r = (a + b - c) / 2
Area of a kite using diagonals d₁ and d₂?
A = ½(d₁ × d₂)
Formula for the measure of one interior angle in a regular pentagon?
[(5 - 2) × 180°] / 5 = 108°
Formula for the measure of one exterior angle in a regular polygon of n sides?
360° / n
Lateral surface area of a right circular cylinder (radius r, height h)?
LA = 2πr × h
Total surface area of a right circular cylinder with radius r, height h?
SA = 2πr² + 2πrh
Volume of a cylinder with radius r, height h?
V = πr²h
Volume of a prism with base area B and height h?
V = B × h
Surface area of a rectangular prism with length ℓ, width w, height h?
SA = 2(ℓw + ℓh + w h)
Volume of a rectangular prism with length ℓ, width w, height h?
V = ℓ × w × h
Volume of a cube with side s?
V = s³
Surface area of a cube with side s?
SA = 6s²
Volume of a cone with radius r, height h?
V = (1/3)πr²h
Lateral surface area of a right circular cone (slant height L, radius r)?
LA = πr × L
Total surface area of a right circular cone with radius r, slant height L?
SA = πrL + πr²
Volume of a pyramid with base area B and height h?
V = (1/3)B × h
Surface area of a regular square pyramid (base side s, slant height L)?
SA = s² + 2sL
Volume of a sphere with radius r?
V = (4/3)πr³
Surface area of a sphere with radius r?
SA = 4πr²
Volume of a hemisphere (radius r)?
V = (2/3)πr³
Surface area of a hemisphere (including the circular base) with radius r?
SA = 3πr²
Formula for the lateral edge length of a regular tetrahedron with side s?
It’s just s (all edges are s). (No special separate formula for an edge.)
Volume of a regular tetrahedron (edge length a)?
V = (a³) / (6√2)
Surface area of a regular tetrahedron with edge a?
SA = √3 × a²
Volume of a triangular prism with base area B and length ℓ?
V = B × ℓ
Area of an isosceles triangle with base b and side lengths s?
A = (b/4) √(4s² - b²)
Area formula for a rhombus using side s and any interior angle θ?
A = s² sin(θ)
Volume of a frustum of a right circular cone, with bases radii r₁, r₂ and height h?
V = (1/3)πh(r₁² + r₁r₂ + r₂²)
Lateral surface area of a frustum of a right circular cone with slant height L and radii r₁, r₂?
LA = π(r₁ + r₂)L
Area of a cyclic quadrilateral (Bretschneider’s formula requires angles, so simpler is Brahmagupta if it’s inscribed, or specifically if it’s a quadrilateral with side lengths a, b, c, d).
Brahmagupta’s formula for cyclic quadrilateral: A = √[(s - a)(s - b)(s - c)(s - d)], s = (a+b+c+d)/2
Radius of the circumscribed circle (circumradius R) for a triangle with sides a, b, c?
R = (a·b·c) / (4A) where A is triangle’s area
Inradius (r) formula of a triangle with semiperimeter s and area A?
r = A / s
Area of a trapezoid in coordinate geometry with vertices (x₁, y₁), (x₂, y₂) horizontally aligned for the parallel bases, etc. – simpler approach: formula for a trapezoid with bases b₁ and b₂, height h?
A = (b₁ + b₂)(h)/2
Formula relating side length s to the diagonal d in a square?
d = s√2
Volume of a wedge or rectangular box cut at an angle can be a complicated scenario – simpler: The volume is still base area × height. For a rectangular-based wedge, the formula is the same as the parallelepiped.
V = area of base × height (No simpler closed form for wedge unless we specify geometry details)
Coordinate geometry formula for the circle with center (h, k) and radius r?
(x - h)² + (y - k)² = r²
Length of a median in a triangle using side lengths – Apollonius’ theorem for median from side a to midpoint?
m_a = ½√(2b² + 2c² - a²)
Area of a parallelogram with vectors 𝑢⃗u and 𝑣⃗v in 2D?
|𝑢⃗u × 𝑣⃗v | (the magnitude of cross product), or in 2D ∣𝑢𝑥𝑣𝑦−𝑢𝑦𝑣𝑥∣
Area of a triangle given coordinates (x₁, y₁), (x₂, y₂), (x₃, y₃)?
A = ½ | x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂) |
Formula for the slope of a line perpendicular to Ax + By = C?
Original slope = -A/B, so perpendicular slope = B/A
Equation of a line passing through (x₀, y₀) perpendicular to Ax + By = C?
B(x - x₀) - A(y - y₀) = 0 (since slope = B/A)
Equation of the angle bisectors in coordinate geometry might be more advanced. Let’s do a simpler one: formula to find the angle θ between two lines with slopes m₁, m₂?
tan θ = |(m₁ - m₂) / (1 + m₁m₂)|
Euler’s formula for planar graphs or polyhedra: V - E + F = 2, typically. That’s topological. Let’s do a simpler geometry one: The length of a segment after a scale factor k about an origin in coordinate geometry?
New length = original length × |k|
Reflection of a point (x, y) across the x-axis results in which formula?
(x, -y)
Reflection of a point (x, y) across the y-axis results in which formula?
(-x, y)
90° rotation about the origin for (x, y) in 2D coordinate geometry?
(-y, x)
180° rotation about the origin for (x, y) in 2D coordinate geometry?
(-x, -y)
Formula for the area of a rectangular coordinate figure from x=a to x=b and y=c to y=d?
Area = (b - a)(d - c)
Area of a parallelogram in coordinate geometry formed by points (x₁, y₁), (x₂, y₂), (x₃, y₃), (x₄, y₄) if it’s a parallelogram? Typically can do cross-product approach. The simpler formula is the absolute value of the cross product of vectors from the same vertex.
A = |(x₂ - x₁)(y₃ - y₁) - (y₂ - y₁)(x₃ - x₁)| if (x₄, y₄) completes the parallelogram
Area of trapezoid (coordinate approach) if top base from (x₁,y₁) to (x₂,y₂) and bottom base from (x₃,y₃) to (x₄,y₄) with bases parallel. Let’s do the simpler standard formula: ½(b₁ + b₂)h.
A = ½ (b₁ + b₂) × height
Arc length formula in degrees for circle radius r and central angle θ in degrees?
Arc length = (θ/360°) × 2πr
Area of a trapezoid with bases b₁ and b₂ and height h?
A = ½ (b₁ + b₂) × height
Sector area formula in degrees for circle radius r and central angle θ in degrees?
Area = (θ/360°) × πr²
Apothem a in a regular n-sided polygon with side length s?
a = (2A) / P
Law of Cosines formula for triangle sides a, b, c, with angle γ opposite side c?
c² = a² + b² - 2ab cos(γ)
Law of Sines formula for triangle with sides a, b, c opposite angles A, B, C respectively?
(a / sinA) = (b / sinB) = (c / sinC)
Surface area formula for a rectangular box with length ℓ, width w, height h?
SA = 2(ℓw + wh + hℓ)
Minimum distance from a point to a plane in 3D given plane Ax + By + Cz + D = 0 and point (x₀, y₀, z₀)?
Distance = |Ax₀ + By₀ + Cz₀ + D| / √(A² + B² + C²)
Volume of a rectangular-based pyramid with base length ℓ, width w, height h?
V = (1/3)(ℓw)h
Area formula for an ellipse with semi-axes a, b?
Area = πab
Measure of each interior angle in a regular dodecagon (12 sides)?
[(12 - 2) × 180°] / 12 = 150°
Altitude (or height) in an equilateral triangle with side s?
h = (s√3)/2
Perimeter of an n-sided regular polygon with side length s?
P = n × s
Segment length on a transversal crossing two parallel lines a distance d apart at an angle θ?
Segment = d / sin(θ)
