GRE - Geometry
What is the formula for the ratio of the circumference C to the diameter d for all circles?
C / d = π. Also can look like C/2r = π This formula is true for all circles π = ratio of the circumference to the diameter in all circles
What is the formula for the circumference of a circle?
C = 2πr
What is the formula for circumference of a circle? What is the formula for area of a circle?
C = 2πr A =πr^2
If a circle has a radius of 5.2, then what is its circumference?
C = 2πr C=2π(5.2) C=10.4π C = 10.4 x 3.14 C = 32.7
Area of a rectangle =
Length x Width
Side opposite the right angle is called the ___. The other 2 sides are ___.
Side opposite the right angle is called the HYPOTENUSE. The other 2 sides are LEGS.
A = 2(πr^2) + 2πrh is the formula for ?
Surface area formula for a right circular cylinder
The hypotenuse of a right triangle is 4 and one leg is 2?
This is a multiple of 1:√3:2 thus 2: 2√3: 4 Note: ratio is ---- leg:leg:hypotenuse
What is the Pythagorean Theorem? Give an example of when it is used? What kind of triangles can this be used on?
a2+b2=c2 can be used on right triangle when 2 sides are known Can be used on 2 special right triangles when 2 sides are known --- isosceles right triangle and 30,60,90 right triangle (this is half of an equilateral triangle)
Angle with measure of less than 90 degrees is called a ____ angle
acute angle
When two parallel lines interstate with a third line... what is true of obtuse and acute angles
acute angles formed are equal obtuse angles formed are equal any acute angle is supplementary to any obtuse angle meaning that they add up to 180 degrees
define a straight angle
an angle whose degree measure is between 0 and 90 degrees
∠ what is this symbol for
angle Example: ∠APC
Picture the sector, chord, segment and arc of a circle.
answer -
What is the perimeter of a right triangle with legs of lengths 3 and 4?
answer: 12 aka (3+4+5) we know this because of special right triangle ratio 3:4:5 Note: ratio is ---- leg:leg:hypotenuse
An entire circle is considered to be an ____ with measure _____ degrees
arc 360 degrees
πr^2 is the formula for what?
area of a circle
A=bh/2 is the formula for
area of a triangle
How you find the area of a sector?
area of sector / area of circle (πr^2) = degree measure of arc/360
Area of sector vs arc length formula
area of sector = (n/360)(area of circle) Arc length = (n/360)(circumference)
The volume of any uniform solid is equal to the _____ times its ____
area of the base times its height aka length x width x height
What are the base and height of a right triangle (used to find area -- area=1/2bh)
base and height of a right triangle are the LEGS Not the hypotensue
Area of a parallelogram =
base x height
Two opposite parralel sides of the trapezoid are called ____ of the trapezoid. How is a trapezoid characterized?
bases of the trapezoid Trapezoid is a quadrilateral in which two sides are parallel (interior angles measure to 360 degrees)
A _____ of a circle is an angle with its vertex at the center of the circle. What is the measure of an arc?
central angle measure of an arc is the measure of its central angle, which is the angle formed by two radii that connect the center of the circle to the two endpoints of the arc
A ____ is a line segment joining two points on a circle.
chord
Any line segment that conjoins two points on a circle is called a ___
chord
diameter is a ____ that passes through the center of the circle
chord chord is any line segment that conjoins two points on a circle
A _____ ______ consists of two bases that are congruent circles lying in parallel planes and a lateral surface made of all line segments that join points on the two circles and that are parallel to the line segment joining the centers of the two circles. The latter line segment is called the axis of the ____.
circular cylinder right circular cylinder is pictured
Distance around a circle is called ____
circumference
2πr is the formula for
circumference of a circle
Circumscribed vs inscribed
circumscribed = shape is on the outside Inscribed = shape is on the inside If a square is circumscribed around a circle it is around the circle if a square is inscribed in circle - it means the square is inside the circle
two angles are ___ if they make up a right angle
complementary angles
Two or more circles with the same center are called ______ circles
concentric circles
triangles are _____ if corresponding angles have the same measure and corresponding sides have the same length
congruent
Angles that have equal measures are called ____
congruent angles opposite or vertical angles have equal measure THUS opposite or vertical angles are congruent angles
Two circles with equal radius are called ____
congruent circles
2 triangles that have the same shape and size are called
congruent triangles
A rectangular solid with six faces is called a ____ in which case l = w = h
cube length = width = height for a cube
volume is expressed in ____ units
cubit units
The vertices of a solid are the points at its corners. A cube has ____ vertices.
eight
How many vertices, edges and faces does a cube have?
eight vertices on a cube twelve edges on a cube rectangle prism (including a cube) has 6 faces
If you see a right angle we can use that to find the ____ in the area formula.
height. use right angles to find height. ** remember height is perpendicular to the base -- aka use the two lines of the triangle that are both touching the right angle -- this means the hypotenuse is not always used when finding the area of a triangle
SSS, SAS and ASA/AAS are used to determine ..?
if two triangles are congruent SSS (side-side-side) -- 3 sides are congruent SAS (side-angle-side) - 2 sides and included angle are congruent ASA (angle side angle) or AAS (angle-angle-side) - 2 angles and included side are congruent
How do you find length of an arc of a circle? What would be the length of an arc if given the radius was 5 and the degree measure of the arc was 50 degrees?
length of an arc to the circumference of a circle ie equal to ratio of the degree measure of the arc to 360 degrees length of arc / circumference = degree of arc/360 length of arc/ 2(5)π = 50/360 solve: answer is 4.4 note = circumference = 2πr
The length of each side of a triangle must be _____ than the sum of lengths of the other two sides
less than Example: 4,7,12 could NOT be lengths of a triangle because 12 is greater than 4+7
If the lengths of two sides of a triangle are unequal, the greater angle lies opposite the ____ side, and vice vera
longer side
A polygon has n sides. To find the number of triangles that can be made out of that polygon the formula ___ can be used ?
n-2
Angle with a measure between 90 degrees and 180 degrees Is a ____ angle
obtuse angle
an angle 90<z<180 is an ___
obtuse angle
In a parallelogram, _____ and ____ are congruent
opposite sides and opposite angles
A quadrilateral in which both pairs of opposite sides are parallel is called a ____
parallelogram
In a rectangle, all angles must be equal to 90°. But for a _____ , no angles need to be equal to 90°.
parallelogram
what does this symbol mean ⊥
perpendicular lines
Two lines that intersect to form 4 congruent angles are called _____
perpendicular lines Each of the 4 angles measures 90 degrees (aka right angles)
A closed figure formed by 3 or more line segments is called a _____
polygon
The ___ states that in a right triangle the hypotenuse is equal to the sum of squares of the length of the legs of the traingle
pythagorean therom
Every quadrilateral has ____ sides and ____ interior angles. What do the interior angles of quadrilateral measure up to?
quadrilateral -- 4 sides + 4 interior angles interior angles add up to 360 degrees
What is the ratio of an isosceles right triangle verses a 30,60,90 right triangle (aka half of an equilateral
ratio of isosceles right triangle = x:x:√2 or 1:1:√2 ratio of 30,60,90 right triangle= x:√3x:2x or 1:√3:2
The Pythagorean theorem can be used on a 30,60, 90 right triangle which is half of an equilateral, what is the ratio of this triangle?
ratio of isosceles right triangle= x:√3x:2x or 1:√3:2
In a _____ the diagonals are equal but in a parallelogram, diagonals are not equal.
rectangle
A quadrilateral with four right angles is called a _____
rectangle -- opposite sides of rectangle are parallel and congruent and two diagonals are also congruent
All ____ are parallelograms
rectangles
Compare edges, vertices and faces of a cube, rectangular pyramid and rectangular prism.
rectangular prism and cube have the same edges and vertices.
A polygon in which all sides are congruent and all interior angles are congruent is called a Give an example
regular polygon octagon = 8 sides sum of measures of interior angle of an octagon is (8-2)(180) = 1080 degrees .... therefore in a regular octagon the measure of each angle is 1080/8 = 135 degrees
what is the ratio of areas between 2 congruent triangles?
remember that congruent triangles have the same size and shape THUS their areas would be equal ratio would be 1:1
An exterior angle of a triangle is equal to the sum of the ______
remote interior angles
An angle with a measure of 90 degrees is called a
right angle
the ____ of a circle is a region bounded by an arc of the circle and two radii
sector
area of a square =
side^2
A rectangular solid is a solid with ____ rectangular faces (all edges meet at ____ angles). examples are cereal boxes, bricks, etc.
six rectangular faces 90 degree angles
___ include those with side ratios: 3:4:5 5:12:13 1:1:√2 1:√3:2 leg:leg:hypotenuse
special right triangles pythagorean theorem can be used for any right triangles Ex: isosceles right triangle, 30,60,90
The ratio of the areas of two similar triangles equals the ____ of the ratio of the corresponding sides
square
a rectangle with four congruent sides is called a ___
square ** type of special quadrilateral (interior angles measure to 360)
What happens when we multiple square roots?
square root x square root cancels out the square root and we are left with the number under the square root
two angles are _____ if the sum of their measures is 180 degrees
supplementary
What is a line that touches only one point on the circumference of the circle. A line drawn tangent to a circle is perpendicular to the radius at at the point of tangency. Line is tangent to a circle O at point T.
tangent
what is formula for area of a circle
πr²
Area of a trapezoid is given by the formula ?
(a+b/2)*h
Area of a trapezoid =
(average of parallel sides)(height) A = ((parallel side + parallel side) /2 ) x height
How can you determine if two triangles are similar triangles?
- vertices can be matched up so corresponding angles are congruent OR Equivalent lengths of corresponding sides have the same ratio -- called scale factor similarity
The length of any side of a triangle is less than the sum of lengths of the other two sides and it is greater than the positive difference of the lengths of the other two sides example: The sides of a triangle are length 5,8 and c. What is the range of possible values of c?
3<C<13 Aka has to be 4-12
A _____ to a circle is a line that lies in the same plane as the circle and intersect exactly at one point. What is this point called?
A TANGENT to a circle is a line that lies in the same plane as the circle and intersect exactly at one point called the POINT OF TANGENCY
Regarding lines and angles ... the sum of measure of all 4 angles =
360 degrees
what are the side ratios of special right triangles?
3:4:5 5:12:13 1:1:√2 (isosceles) 1:√3:2 (30,60,90) Note: ratio is ---- leg:leg:hypotenuse
what are the quadrants on a graph
I, II, III, and IV
What are the bases of a trapezoid?
two lines that are opposite and parallel note - trapezoid is characterized by two opposite parallel lines (trapezoid is a quadrilateral)
What is a convex polygon?
a polygon in which each interior angle has a measure less than 180
If a polygon has 5 sides. How many triangles can be made out of it?
use formula n-2 5-2 = 3 pentagon (5 sided polygon) has 3 triangles that can be made out of it
slope-intercept equation equals
y=mx+b m = slope b = y intercept
We know that we can use (n-2) to find number of triangles in a polygon (n = sides of polygon). We also know that the sum of interior angles of a triangle = 180 degrees. THUS.... the sum of measures of interior angles of an n-sided polygon is _____ (give formula)
(n-2)(180)
If a hexagon has 6 sides, what is the sum of measures of interior angles of the hexagon?
(n-2)(180) (6-2)(180) 4x180 =720 n = sides of polygon (n-2) = number of triangles in a polygon 180 degrees = sum of angles in triangle
What is the formula for area of a sector?
(n/360)(area of circle) n= central angle measure area of circle = πr^2
what is the formula to find arc length of a circle?
(n/360)(circumference of circle) n = central angle (in degrees) circumference =2πr
what is the hypotenuse of a right triangle if the legs are 12 and 16?
12:16:? This is a multiple of 3:4:5 Thus 12:16:20 Note: ratio is leg:leg:hypotenuse
Sum on interior angles of a triangle are
180 degrees
The ratio of an isosceles right triangle is x:x:√2 what could this also be written as?
1:1:√2 = x:x:√2
An isosceles triangle has ___ congruent sides.
2 congruent sides the angles opposite the 2 congruent sides are equal.
2 triangles that have the same shape and size are called ____ 2 triangles that have the same shape but not necessarily the same size are ____
2 triangles that have the same shape and size are called CONGRUENT TRIANGLES 2 triangles that have the same shape but not necessarily the same size are SIMILAR TRIANGLES
diameter of circle =
2r 2 times the radius of the circle
An equilateral triangle has ______ congruent sides. What is the measure of each interior angle?
3 congruent sides 60 degrees = each interior angle (180 = total)
What is the value of π? What can it be approximated to as a fraction?
3.14 22/7
Common on GRE -- If you see a triangle with legs with lengths of 3 and 4. What is the length of the. hypotenuse
5 Pythagorean theorem in addition any multiple of these lengths makes another pythagorean triple for instance 6^2 + 8^2 = 10^2 another triple seen frequently is 5,12 and 13
A pentagon is a polygon with ___ sides. how many triangles can be made out of a pentagon?
5 sides 3 triangles
Angle x is an exterior angle of a triangle with remote interior angles 50 and 80 degrees. What is the value of x in degrees?
50+80=130 degrees answer: 130 degrees Note that an exterior angle of a triangle is equal to the sum of the remote interior angles
What is the perimeter of a triangle with lengths 5,6,8
= 19 add all sides together to find perimeter of triangle
What is the surface area formula for a right circular cylinder?
A = 2(πr^2) + 2πrh ** sum of areas of the two bases and the area of its lateral surface
For all parallelograms, including rectangles and squares the area A is given by the formula --?
A = bh b= length of base h = length of corresponding height A = area
What is area of a parallelogram
A = bh same as area of rectangle and squares
What is the area of a triangle?
A = bh/2
For similar triangles, the ratio of their corresponding sides is 2:3. What is the ratio of their areas?
Answer = 2^2:3^2 Answer = 4:9 The ratio of the areas of two similar triangles equals the square of the ratio of the corresponding sides
The ratio of the areas of two similar triangles is the square of the ratio of corresponding lengths. For example: triangle ABC = 2:3:4 and we have a similar triangle DEF which ratio is 4:6:8) To find the ratio of area's between the two triangles we use the ratio of corresponding lengths squared (in this case 2 - because DEF is 2x greater than ABC) In this case, what would the ratio of areas between the two triangles be?
Area of DEF/ Area of ABC = DE/AB = (2/1)^2 = 4
Area of rectangular prism = Surface area of a rectangular prism =
Area of rectangular prism = l x w x h Surface area of a rectangular prism = 2(lw x lh x 2wh)
The hypotenuse of an isosceles right triangle is 16. What is the area of the triangle.
Important things to note: base and height of a right triangle are the LEGS ratio of isosceles right triangle = 1:1:√2 we know the hypotenuse is 16 --- we know that the two legs isosceles triangle are equal to each other To get the length of legs we divide 16/√2 --- think about why we do this. If we had an isosceles right triangle with a hypotenuse of √2 we could divide this by √2 to get the length of the legs 1. simplify 16/√2 by multiplying top and bottom by √2. 16√2 / √2 *√2 = 16√2 /2 = 8/√2 We now know legs of isoclese right triangle are 8√2 A = 1/2*8√2*8√2 Remember square root x square root cancels out the square root and we are left with the number under the square root A =1/2(64)(2) A= 64
What is the hypotenuse of a right triangle if the legs are 5 and 5?
Note that this is a multiple of 1:1:√2 (isosceles right triangle) THUS, the hypotenuse would be 5√2 Note: ratio is leg:leg:hypotenuse
A square has a diagonal of length of 5. What is the length of a side of the square???
Note: The diagonal of a square is the hypotenuse of an isosceles right triangle whose legs are sides the of the square. Thus... 5 is the hypotenuse of an isosceles right triangle. We know 1:1:√2 To find the lengths of the other legs we must divide our hypotenuse by √2 (think about why we do this. If we had an isosceles right triangle with a hypotenuse of √2 we could divide this by √2 to get the length of the legs 1.) Thus 5/√2 --> simplify by multiple top and bottom by √2 which equals 5√2/2 length of legs = 5√2/2 ** remember the rule of multiple roots (ex: √2 x √2 = 2)
Regular pentagon ABCDE has a side length of 28. Regular pentagon FGHIJ has a perimeter of 60 and an area of about 248. What is the ratio of the are of ABCDE to the area of FGHIJ?
Note: because it says that the figures are "regular" we know that the sides are all the same. To find the measure of a side of regular pentagon FGHIJ, we can divide the perimeter (60) by the number of sides on a pentagon (5). 60/5 = 12 We now know the side length of one pentagon is 12 and one is 28. Thus the ratio would be 28:12 --> sampled to 7:3. Now to find the area of ratios between to similar pentagons we square the ratio of sides. 7^2:3^2 = 49:9 49.9 is the ratio of areas
A sector of a circle is a region bounded by an arc of the circle and two radii. The ratio of the area of a sector of a circle to the area of the entire circle is equal to the ratio of the degree measure of the arc to 360. Remember area formula for a circle is A= πr^2 If given the degree of arc ABC is 50 degrees and the radius of the circle is r =5 what would the formula look like. Let S denote area of the sector of a circle
S/(5^2)π = 50/360 S/25π = 5/360 area of sector = 10.9
The surface are of a rectangular solid is the sum of the areas of the six faces or ... formula?
SA = 2(lw +lh +wh)
Congruent triangles are triangles that have the same shape and size. There are 3 propositions for determine whether 2 triangles are congruent. What are they?
SSS (side-side-side) -- 3 sides are congruent SAS (side-angle-side) - 2 sides and included angle are congruent ASA (angle side angle) or AAS (angle-angle-side) - 2 angles and included side are congruent
volume of cube can be found by?
S^3 side length to the third
What is the volume of a rectangular solid?
V = lwh
what is the formula for volume of a right circular cylinder
V = πr^2h
If a circular cylinder has a height of 6.5 and a base with a radius of 3 what is the volume? What is the surface area?
V = πr^2h V =π(3^2)(6.5) V = 58.5π A = 2(πr^2) + 2πrh A = 2(π3^2) + 2π(3)(6.5) A = 2(9π) + 2π(19.5) A = 18π + 39π A = 57π
What is the volume of a cylinder? What is the surface area of a cylinder?
V=πr^2h Surface area = (2πr^2) + 2πrh --- SA explained: 2 x area of base + are of rest of shell (aka bxh but in cylinder the base is found by finding circumference thus it is circumference x height) note circumference of a circle = 2πr
V = πr^2h what formula is this
Volume of a right circular cylinder
Important things to note about cubes -- each edge is equal in length -- all faces are squares -- edge of a cube is commonly represented by variable e What is the volume of a cube? What is the SA of cube?
Volume of cube = l x w x h (edges ^ 3 = e^3) SA = Sum of areas of faces -- 6e^2
The edges of a solid are the line segments that connect the vertices and form the sides of each face of the solid. fore example a cube has_____ edges The vertices of a solid are the points at its corners. A cube has ____ vertices. The faces of a solid are the polygons that form the outside of the solid. A rectangular has ____ faces, all rectangles. A cube (which is a rectangular prism has ____ faces, all squares)
eight vertices on a cube twelve edges on a cube rectangle prism (including a cube) has 6 faces
opposite angles or vertical angles have _____
equal measures
An ____ triangle has three sides of equal length and three 60 degree angles
equilateral triangle
What are 3 types of special triangles?
equilateral, isosceles and right triangle
2(lw +lh +wh) what is this formula for?
finding surface area of rectangular solid which is the sum of the six faces
A quadrilateral is a polygon with ___ sides. how many triangles can be made out of a quadrilateral?
four (two triangles can be made out of a quadrilateral)
rectangle, parallelogram, square and trapezoid are all ___
four special types of quadrilaterals
A polygon is ________ in a circle if all its vertices lie on the circle or equivalently the circle is circumscribed about the polygon. If one side of an inside triangle is a diameter of the circle, then triangle is a right triangle. Conversely, if an inscribed triangle is a right triangle, then one of its sides is a diameter of the circle.
inscribed
The diagonal of a square is the hypotenuse of an __________ whose legs are the sides of the square.
isosceles right triangle
what is a special right triangle with ratio of its side 1:1:√2
isosceles right triangle
The pythagorean theorem can be used on a ________ right triangle where 2 congruent sides + angle opposite two congruent sides are equal.
isosceles right triangle ** will have two 45 degree angles and one 90 degree angle adjacent from the hypotenuse
A quadrilateral in which one pair of opposite sides is parallel is called a _____
trapezoid
A polygon that has 3 sides, 3 angles and measure of interior angles adds up to 180
triangle
What is the simplest polygon?
triangle (3 sides)
A ____ of a polygon is the point where two adjacent sides meet
vertex
point of interaction is called the ___ of an angle
vertex of an angle
the endpoints of polygons are called
vertices
volume of a rectangular solid can be found by
volume = length x width x height
If a rectangular solid has length 8.5, width 5, and height 10 then what is its volume? What about surface area?
volume of rectangular solid= l*w*h volume = (8.5x5x10) = 425 surface area of rectangular solid= 2(lw*lh*wh) surface area = 2((8.5*5)(8.5*10)(10*5)) surface area = 355
what is the scale factor of similarity? what can this be used for?
when lengths of corresponding sides have the same ratio this can be used to determine if triangles are similar?