Geometry Basic Flashcards
1
Q
Sum of angles on a straight line
A
180°
2
Q
Sum of angles around a point
A
360°
3
Q
Definition of a right angle
A
90°
4
Q
Complementary angles sum to
A
90°
5
Q
Supplementary angles sum to
A
180°
6
Q
Two lines that never intersect (in Euclidean geometry)
A
Parallel lines
7
Q
Angles across from each other when two lines intersect
A
Vertical angles
8
Q
Sum of vertical angles
A
They are equal (same measure)
9
Q
Alternate interior angles when lines are parallel are
A
Equal
10
Q
Definition of a transversal line
A
A line that intersects two or more other lines
11
Q
Sum of interior angles in a triangle
A
180°
12
Q
An isosceles triangle has how many equal sides?
A
Two
13
Q
An equilateral triangle has angles measuring
A
60° each
14
Q
A right triangle has one angle measuring
A
90°
15
Q
Pythagorean theorem formula
A
a² + b² = c²
16
Q
Type of triangle with all angles < 90°
A
Acute triangle
17
Q
Type of triangle with one angle > 90°
A
Obtuse triangle
18
Q
Median of a triangle connects a vertex to
A
The midpoint of the opposite side
19
Q
Altitude (height) of a triangle is drawn
A
Perpendicular to the opposite side
20
Q
Sum of two sides of a triangle is always
A
Greater than the third side
21
Q
Sum of interior angles in a quadrilateral
A
360°
22
Q
Rectangle definition
A
A quadrilateral with four right angles
23
Q
Square definition
A
A rectangle with all sides equal
24
Q
Parallelogram definition
A
Opposite sides parallel and equal
25
Rhombus definition
A parallelogram with four equal sides
26
A trapezoid (US) or trapezium (UK) has
At least one pair of parallel sides
27
Kite definition (in geometry)
A quadrilateral with two pairs of adjacent equal sides
28
Opposite angles in a parallelogram are
Equal
29
Diagonals of a rectangle are
Equal in length
30
Diagonals of a parallelogram
Bisect each other
31
A pentagon has how many sides?
5
32
A hexagon has how many sides?
6
33
A heptagon (or septagon) has how many sides?
7
34
Sum of interior angles of a pentagon
540°
35
Sum of interior angles of a hexagon
720°
36
Formula for sum of interior angles of an n-sided polygon
(n - 2) × 180°
37
A regular polygon has
All sides and angles equal
38
Number of diagonals in an n-sided polygon
n(n - 3)/2
39
Interior angle of a regular quadrilateral (square)
90°
40
Interior angle of a regular triangle (equilateral)
60°
41
Circumference formula
C = 2πr
42
Area of a circle formula
A = πr²
43
Diameter of a circle
2r
44
Definition of a chord
Line segment joining two points on a circle
45
Definition of a secant
Line that intersects a circle in two points
46
Definition of a tangent to a circle
Line that touches the circle at exactly one point
47
Tangents from a common external point to a circle are
Equal in length
48
Definition of a central angle
An angle whose vertex is the center of the circle
49
Definition of an inscribed angle
An angle formed by two chords with the vertex on the circle
50
Inscribed angle theorem
An inscribed angle is half the measure of its intercepted arc
51
Slope formula between (x₁, y₁) and (x₂, y₂)
(y₂ - y₁) / (x₂ - x₁)
52
Midpoint formula
((x₁ + x₂)/2, (y₁ + y₂)/2)
53
Distance formula
√[(x₂ - x₁)² + (y₂ - y₁)²]
54
Equation of a line in slope-intercept form
y = mx + b
55
Standard form of a line
Ax + By = C
56
Slope of x-axis
0
57
Slope of y-axis
Undefined (vertical line)
58
Point-slope form of a line
y - y₁ = m(x - x₁)
59
Length of the vector (a, b)
√(a² + b²)
60
Equation of a circle in center-radius form
(x - h)² + (y - k)² = r²
61
Translation moves a shape
Without rotation; shifts every point by the same distance
62
Reflection definition
A flip over a line or axis producing a mirror image
63
Rotation definition
Turning a figure about a fixed point (the center of rotation)
64
90° rotation about the origin (x, y)
(-y, x)
65
180° rotation about the origin (x, y)
(-x, -y)
66
Dilation definition
Enlarges or reduces a figure using a scale factor
67
Scale factor = 2 means
All distances double from the center of dilation
68
Invariant point under reflection across x-axis
A point on the x-axis (y=0)
69
Invariant point under reflection across y-axis
A point on the y-axis (x=0)
70
Composition of transformations
Applying multiple transformations sequentially
71
Cube definition
A prism with six square faces
72
Rectangular prism definition
A prism with rectangular faces
73
Cylinder definition
A solid with two congruent circular bases connected by a curved surface
74
Sphere definition
A set of points in 3D at a fixed distance (radius) from a center
75
Cone definition
A solid with a circular base and a vertex (apex) not in the plane of the circle
76
Pyramid definition
A solid with a polygon base and triangular faces meeting at an apex
77
Tetrahedron definition
A pyramid with a triangular base (4 triangular faces)
78
Edges in a cube
12
79
Faces in a rectangular prism
6
80
Euler’s formula for polyhedra
V - E + F = 2
81
Perimeter of a square with side s
4s
82
Perimeter of a rectangle with length L and width W
2(L + W)
83
Area of a square with side s
s²
84
Area of a rectangle with length L and width W
LW
85
Area of a parallelogram
Base × Height
86
Area of a triangle
½ (Base × Height)
87
Surface area of a cube with side s
6s²
88
Volume of a cube with side s
s³
89
Volume of a rectangular prism with length L, width W, height H
LWH
90
Volume of a cylinder with radius r and height h
πr²h
91
Volume of a cone
(1/3)πr²h
92
Volume of a sphere
(4/3)πr³
93
Surface area of a sphere
4πr²
94
Diagonals of a rectangle
Are equal and bisect each other
95
Inradius of a right triangle with legs a, b, and hypotenuse c
(a + b - c)/2
96
Triangle inequality property
Sum of any two sides > third side
97
Exterior angle of a triangle equals
Sum of the two non-adjacent interior angles
98
Altitude to the hypotenuse in a right triangle forms
Two similar subtriangles to the original
99
Midsegment of a triangle (connecting midpoints of two sides)
Parallel to the third side and half its length
100
Lateral surface area of a cylinder
2πrh
101
Define a line segment in geometry.
A portion of a line bounded by two distinct endpoints.
102
Define a ray in geometry.
A part of a line starting at one endpoint and extending infinitely in one direction.
103
What is a transversal line?
A line that intersects two or more lines at distinct points.
104
What does it mean for two angles to be adjacent?
They share a common vertex and a common side but do not overlap.
105
Define a linear pair of angles.
Two adjacent angles whose non-common sides form a straight line (sum is 180°).
106
What is a convex polygon?
A polygon with all interior angles less than 180°, and vertices 'point outward.'
107
What is a concave polygon?
A polygon with at least one interior angle greater than 180°, and a ‘dent’ in its shape.
108
Define the term ‘diagonal’ in a polygon.
A line segment connecting two non-adjacent vertices.
109
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.
110
Define an exterior angle of a polygon.
An angle formed by extending one side of the polygon at a vertex.
111
State the Exterior Angle Theorem (for triangles).
An exterior angle of a triangle equals the sum of the two non-adjacent interior angles.
112
What is the centroid of a triangle?
The point where the three medians intersect (it divides each median in a 2:1 ratio).
113
Define the incenter of a triangle.
The point where the three angle bisectors intersect; center of the inscribed circle.
114
Define the circumcenter of a triangle.
The point where the perpendicular bisectors of the sides intersect; center of the circumscribed circle.
115
Define the orthocenter of a triangle.
The point where the three altitudes intersect.
116
State the formula for the perimeter of an equilateral triangle with side s.
P = 3s.
117
Define the concept of similar triangles.
Triangles that have the same shape but not necessarily the same size (corresponding angles equal, sides in proportion).
118
State the Triangle Proportionality Theorem.
A line parallel to one side of a triangle divides the other two sides proportionally.
119
What is the formula for the area of an isosceles triangle with base b and height h?
A = (1/2) × b × h.
120
Define the circumscribed circle of a triangle.
A circle passing through all three vertices of the triangle.
121
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.
122
Define the altitude of a trapezoid.
The perpendicular distance between the two parallel sides.
123
What is a regular polygon?
A polygon with all sides and angles equal (e.g., regular pentagon, regular hexagon).
124
Define an n-sided polygon’s interior angle measure if it is regular.
Each interior angle = [(n - 2) × 180°] / n.
125
State the formula for finding the total number of diagonals in an n-sided polygon.
[n(n - 3)] / 2.
126
Define a central angle in a regular polygon.
An angle formed by two radii drawn to consecutive vertices of the polygon.
127
What is the measure of each central angle in a regular n-sided polygon?
360° / n.
128
Define the apothem of a regular polygon.
A line from the center perpendicular to any side, used in finding area.
129
Write the area formula for a regular polygon with perimeter p and apothem a.
Area = (1/2) × p × a.
130
Define a frustum of a cone.
The shape obtained by slicing a cone parallel to its base and removing the top portion.
131
Define an arc of a circle.
A portion of the circumference between two points on the circle.
132
What is the measure of a minor arc?
Equal to the measure of the central angle that subtends it, in degrees.
133
What is the formula for arc length of a circle with radius r and subtended angle θ (in degrees)?
Arc length = (θ/360°) × 2πr.
134
Define a major arc in a circle.
An arc larger than a semicircle, more than 180°.
135
Define a semicircle in a circle.
An arc that measures exactly 180°.
136
What is a chord’s perpendicular bisector property in a circle?
It passes through the center of the circle.
137
Explain the inscribed angle theorem in circles.
The measure of an inscribed angle is half the measure of its intercepted arc.
138
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.
139
Define a segment of a circle (minor segment).
A region formed by a chord and the arc it subtends.
140
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.
141
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.
142
What is the tangent-secant theorem?
Tangent² = External segment × Whole secant (if a secant is drawn from the same external point).
143
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).
144
Define two tangents from an external point to a circle.
They are congruent (equal in length).
145
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.
146
Define concentric circles.
Circles with the same center but different radii.
147
Write the distance formula for two points (x₁, y₁) and (x₂, y₂) in the plane.
√[(x₂ - x₁)² + (y₂ - y₁)²].
148
What is the midpoint formula in the Cartesian plane for points (x₁, y₁) and (x₂, y₂)?
((x₁ + x₂)/2, (y₁ + y₂)/2).
149
Define the slope of a line in coordinate geometry.
(y₂ - y₁) / (x₂ - x₁), for x₂ ≠ x₁.
150
Write the equation of a circle with center (h, k) and radius r.
(x - h)² + (y - k)² = r².
151
Define the equation of a horizontal line through (x₀, y₀).
y = y₀.
152
Define the equation of a vertical line through (x₀, y₀).
x = x₀.
153
What is the formula for the sum of the interior angles of an n-sided polygon?
(n - 2) × 180°
154
State the formula for the measure of each interior angle in a regular n-sided polygon.
[(n - 2) × 180°] / n
155
What is the perimeter formula for a regular n-sided polygon with side length s?
P = n × s
156
Give the formula for the sum of exterior angles of any convex polygon.
360°
157
State the formula for the measure of each exterior angle of a regular n-sided polygon.
360° / n
158
Formula for the slope m between two points (x₁, y₁) and (x₂, y₂) in the plane?
m = (y₂ - y₁) / (x₂ - x₁)
159
Write the distance formula between (x₁, y₁) and (x₂, y₂) in 2D coordinate geometry.
d = √[(x₂ - x₁)² + (y₂ - y₁)²]
160
Give the midpoint formula between (x₁, y₁) and (x₂, y₂).
((x₁ + x₂)/2, (y₁ + y₂)/2)
161
Formula for the equation of a line in point-slope form?
y - y₁ = m(x - x₁)
162
Formula for the equation of a line in slope-intercept form?
y = mx + b
163
Standard form of a line equation in the plane?
Ax + By = C
164
Circumference formula for a circle of radius r?
C = 2πr
165
Area formula for a circle of radius r?
A = πr²
166
Arc length formula for a circle with radius r subtended by central angle θ (in radians)?
Arc length = rθ
167
Area of a sector formula for angle θ (in radians) in a circle of radius r?
Area of sector = (θ/2)r²
168
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)
169
Area of an ellipse with semi-major axis a and semi-minor axis b?
A = πab
170
Distance formula from a point (x₀, y₀) to the line Ax + By + C = 0?
Distance = |Ax₀ + By₀ + C| / √(A² + B²)
171
Formula for the slope of a perpendicular line to y = m₁x + b₁?
m₂ = -1/m₁
172
Sum of the interior angles of a triangle?
180°
173
Perimeter formula for a triangle with sides a, b, c?
P = a + b + c
174
Formula for the area of a triangle given base b and height h?
A = ½ × b × h
175
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
176
Formula for area of an equilateral triangle with side s?
A = (√3 / 4) s²
177
Pythagorean theorem for a right triangle with legs a, b and hypotenuse c?
a² + b² = c²
178
Perimeter of a rectangle with length ℓ and width w?
P = 2(ℓ + w)
179
Area of a rectangle with length ℓ and width w?
A = ℓ × w
180
Formula for the diagonal of a rectangle with length ℓ and width w?
d = √(ℓ² + w²)
181
Perimeter of a square with side s?
P = 4s
182
Area of a square with side s?
A = s²
183
Perimeter of a parallelogram with sides a, b?
P = 2(a + b)
184
Area of a parallelogram with base b and height h?
A = b × h
185
Perimeter of a rhombus with side s?
P = 4s
186
Area of a rhombus using diagonals d₁ and d₂?
A = ½(d₁ × d₂)
187
Area of a trapezoid (US) or trapezium (UK) with bases b₁, b₂ and height h?
A = ½(b₁ + b₂) × h
188
Sum of the interior angles of a quadrilateral?
360°
189
Inradius (r) formula for a right triangle with legs a, b and hypotenuse c?
r = (a + b - c) / 2
190
Area of a kite using diagonals d₁ and d₂?
A = ½(d₁ × d₂)
191
Formula for the measure of one interior angle in a regular pentagon?
[(5 - 2) × 180°] / 5 = 108°
192
Formula for the measure of one exterior angle in a regular polygon of n sides?
360° / n
193
Lateral surface area of a right circular cylinder (radius r, height h)?
LA = 2πr × h
194
Total surface area of a right circular cylinder with radius r, height h?
SA = 2πr² + 2πrh
195
Volume of a cylinder with radius r, height h?
V = πr²h
196
Volume of a prism with base area B and height h?
V = B × h
197
Surface area of a rectangular prism with length ℓ, width w, height h?
SA = 2(ℓw + ℓh + w h)
198
Volume of a rectangular prism with length ℓ, width w, height h?
V = ℓ × w × h
199
Volume of a cube with side s?
V = s³
200
Surface area of a cube with side s?
SA = 6s²
201
Volume of a cone with radius r, height h?
V = (1/3)πr²h
202
Lateral surface area of a right circular cone (slant height L, radius r)?
LA = πr × L
203
Total surface area of a right circular cone with radius r, slant height L?
SA = πrL + πr²
204
Volume of a pyramid with base area B and height h?
V = (1/3)B × h
205
Surface area of a regular square pyramid (base side s, slant height L)?
SA = s² + 2sL
206
Volume of a sphere with radius r?
V = (4/3)πr³
207
Surface area of a sphere with radius r?
SA = 4πr²
208
Volume of a hemisphere (radius r)?
V = (2/3)πr³
209
Surface area of a hemisphere (including the circular base) with radius r?
SA = 3πr²
210
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.)
211
Volume of a regular tetrahedron (edge length a)?
V = (a³) / (6√2)
212
Surface area of a regular tetrahedron with edge a?
SA = √3 × a²
213
Volume of a triangular prism with base area B and length ℓ?
V = B × ℓ
214
Area of an isosceles triangle with base b and side lengths s?
A = (b/4) √(4s² - b²)
215
Area formula for a rhombus using side s and any interior angle θ?
A = s² sin(θ)
216
Volume of a frustum of a right circular cone, with bases radii r₁, r₂ and height h?
V = (1/3)πh(r₁² + r₁r₂ + r₂²)
217
Lateral surface area of a frustum of a right circular cone with slant height L and radii r₁, r₂?
LA = π(r₁ + r₂)L
218
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
219
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
220
Inradius (r) formula of a triangle with semiperimeter s and area A?
r = A / s
221
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
222
Formula relating side length s to the diagonal d in a square?
d = s√2
223
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)
224
Coordinate geometry formula for the circle with center (h, k) and radius r?
(x - h)² + (y - k)² = r²
225
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)
226
Area of a parallelogram with vectors 𝑢⃗u and 𝑣⃗v in 2D?
|𝑢⃗u × 𝑣⃗v | (the magnitude of cross product), or in 2D ∣𝑢𝑥𝑣𝑦−𝑢𝑦𝑣𝑥∣
227
Area of a triangle given coordinates (x₁, y₁), (x₂, y₂), (x₃, y₃)?
A = ½ | x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂) |
228
Formula for the slope of a line perpendicular to Ax + By = C?
Original slope = -A/B, so perpendicular slope = B/A
229
Equation of a line passing through (x₀, y₀) perpendicular to Ax + By = C?
B(x - x₀) - A(y - y₀) = 0 (since slope = B/A)
230
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)|
231
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|
232
Reflection of a point (x, y) across the x-axis results in which formula?
(x, -y)
233
Reflection of a point (x, y) across the y-axis results in which formula?
(-x, y)
234
90° rotation about the origin for (x, y) in 2D coordinate geometry?
(-y, x)
235
180° rotation about the origin for (x, y) in 2D coordinate geometry?
(-x, -y)
236
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)
237
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
238
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
239
Arc length formula in degrees for circle radius r and central angle θ in degrees?
Arc length = (θ/360°) × 2πr
240
Area of a trapezoid with bases b₁ and b₂ and height h?
A = ½ (b₁ + b₂) × height
241
Sector area formula in degrees for circle radius r and central angle θ in degrees?
Area = (θ/360°) × πr²
242
Apothem a in a regular n-sided polygon with side length s?
a = (2A) / P
243
Law of Cosines formula for triangle sides a, b, c, with angle γ opposite side c?
c² = a² + b² - 2ab cos(γ)
244
Law of Sines formula for triangle with sides a, b, c opposite angles A, B, C respectively?
(a / sinA) = (b / sinB) = (c / sinC)
245
Surface area formula for a rectangular box with length ℓ, width w, height h?
SA = 2(ℓw + wh + hℓ)
246
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)
247
Volume of a rectangular-based pyramid with base length ℓ, width w, height h?
V = (1/3)(ℓw)h
248
Area formula for an ellipse with semi-axes a, b?
Area = πab
249
Measure of each interior angle in a regular dodecagon (12 sides)?
[(12 - 2) × 180°] / 12 = 150°
250
Altitude (or height) in an equilateral triangle with side s?
h = (s√3)/2
251
Perimeter of an n-sided regular polygon with side length s?
P = n × s
252
Segment length on a transversal crossing two parallel lines a distance d apart at an angle θ?
Segment = d / sin(θ)
