##### Figures - revision

To understand what geometry is, you need to know basic informations about figures so we prepared for you quick revision:

- ###### Triangle

  is a polygon with three edges and three vertices.

- ###### Angles

  sum of all three angles is equal 180 degrees.

- ###### Right triangle

  is a triangle in which one angle is a right angle (90-degree angle).

- ###### Right triangle

  has 2 perpendicular (sides adjacent to the right angle) and 1 hypotenuse (side in opposite to right angle)

- ###### Isosceles triangle

  is a triangle that has two sides of equal length and two angles that are also equal.

  ###### Important

  The third side of the isosceles triangle is usually called the base. The two equal sides are called legs. In isosceles triangle two angles next to base are equal to each other

- ###### Equilateral triangle

  is a type of triangle that has all three sides equal and angles also equal to each other, measuring 60 degrees each.

- ###### Pythagorean Theorem

  In a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

- ###### Square

  is a shape with four straight sides of equal length and four right angles (90 degree angles). It is a type of rectangle where all sides are the same length.

  ###### Warning

  in every four-sided shape the sum of angles is equal 360 degrees.

  ###### Important

  the straight line between two opposite corners is diagonal. In square the formula for diagonal is: d = a√2 (where "d" is diagonal, "a" is a side of the square)

- ###### Rectangle

  is a shape with four straight sides and four right angles (90 degree angles), just like a square. However, unlike a square, a rectangle has two pairs of sides with equal length, but opposite sides are different lengths.square is a type of rectangle, not all rectangles are squares.

  ###### Warning

  square is a type of rectangle, but not all rectangles are squares.

  ###### Important

  As you now know, the straight line between two opposite corners is diagonal. If you want to calculate the diagonal, then you have to use and transform "Pythagorean Theorem", so d = √(a² + b²)

- ###### Rhombus

  is another four-sided shape with four equal-length sides, like a square. However, a rhombus does not have right angles - instead, its opposite angles are equal to each other.

  ###### Warning

  a square is a type of rhombus, but not all rhombuses are squares, and they have a different shape overall.

- ###### Parallelogram

  is a four-sided shape with opposite sides that are parallel to each other. This means that the opposite sides never meet or intersect. Like a rectangle and a rhombus, a parallelogram also has four right angles.

  ###### Warning

  its sides do not have to be the same length like in a square or rhombus. So while a square and a rhombus are types of parallelograms, not all parallelograms are squares or rhombuses.

  ###### Warning

  Opposite interior angles of a parallelogram are congruent, which means they have the same measure.

- ###### Trapezium

  is a quadrilateral with at least one pair of parallel sides. The parallel sides of a trapezium are called the bases, and the non-parallel sides are called the legs. In a trapezium, the angles on the same side of the parallel sides are supplementary, which means they add up to 180 degrees.

- ###### Circle

  is a two-dimensional shape that is defined by a curved line that is always the same distance from a point called the center. The distance from the center to any point on the circle is called the radius, and the distance across the circle, passing through the center and touching two points on the circumference, is called the diameter.

  ###### Important

  The formula for circumference of circle is: c = 2πr (where "r" is the radius of the circle, and "π" (pi) is a mathematical constant approximately equal to 3.14159.)

  ###### Important

  The formula for area (A) of circle is: A = πr²

##### Pythagorean Theorem

In a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

##### Types of lines

Trygonometry is a branch of mathematics between angles and ratios of lengths in triangles. There are 4 trigonometric functions, such as:

Line

###### Line

is a one-dimensional geometric figure that extends infinitely in both directions.

###### Ray

is a one-dimensional geometric object that consists of a starting point (called the endpoint) and extends infinitely in one direction.

###### Segment

is a one-dimensional geometric object that consists of two endpoints and all the points on the straight line between those two endpoints.

###### Intersecting lines

are two or more lines that share at least one point in common. The point at which the lines intersect is called the point of intersection.

###### Parallel lines

are two or more lines that are always the same distance apart and never intersect, no matter how far they are extended. Parallel lines have the same slope, and they remain equidistant from each other at all points.

###### Perpendicular lines

are two lines that intersect at a right angle (90 degrees). This means that the two lines form four right angles at their point of intersection.

##### Cartesian coordinate system

Is a coordinate system in which points in a plane are located by their distances from two perpendicular lines, usually called the x-axis and the y-axis.

In the Cartesian coordinate system, each point in the plane is uniquely determined by its x-coordinate and its y-coordinate. The x-coordinate is the horizontal distance from the origin, which is the point where the x-axis and the y-axis intersect, to the point in question. The y-coordinate is the vertical distance from the origin to the point in question.

##### Functions

Function is a rule that assigns a unique output value for every input value. It can be thought of as a machine that takes in input values, performs some operations on them, and produces an output value. The input values are typically represented by the variable "x", and the output values are represented by the variable "y". The function is often denoted by the symbol "f(x)" or "g(x)", where "x" is the input variable and "f" or "g" is the name of the function. The basic idea is that you can use a function to describe a relationship between two quantities, and to make predictions about how they will behave in different situations.

##### Linear function

Linear function is a specific type of function in mathematics that has the form "f(x) = ax + b", where "a" and "b" are constants. In a linear function, the input variable "x" appears with a power of 1, and the output variable "f(x)" is a straight line on a graph. The constant "a" represents the slope of the line, which is how steeply it rises or falls, and the constant "b" represents the y-intercept of the line, which is where the line intersects the y-axis.

###### Example:

##### Quadratic function

Quadratic function is a type of function in mathematics that has the form "f(x) = ax² + bx + c", where "a", "b", and "c" are constants. In a quadratic function, the input variable "x" appears with a power of 2, and the output variable "f(x)" is a curve called a parabola when graphed. The constant "a" determines the shape of the parabola and whether it opens upwards or downwards, the constant "b" determines the horizontal position of the vertex (the highest or lowest point on the parabola), and the constant "c" determines the vertical position of the vertex (the height of the vertex above the x-axis).

###### Example:

##### Exponential function

Exponential function is a mathematical function of the form f(x) = aˣ, where "a" is a positive constant (known as the base of the function) and "x" is a variable. The value of the function at any input value of "x" is obtained by raising the base "a" to the power of "x".

###### Example:

##### Logarithmic function

Logarithmic function is a mathematical function that represents the inverse of an exponential function. In other words, if y = aˣ is an exponential function, then the logarithmic function with base a is defined as y = logₐ(x), where logₐ(x) is the exponent to which b must be raised to obtain x.

###### Example:

##### Rational function

Rational function is a type of mathematical function that can be expressed as the ratio of two polynomial functions, where the denominator is not equal to zero. In other words, it's a function of the form f(x) = p(x) / q(x), where p(x) and q(x) are polynomial functions and q(x) is not equal to zero for any value of x.

###### Example:

##### Trigonometric function

Trigonometric function is a mathematical function that relates the angles of a right-angled triangle to the lengths of its sides. There are six trigonometric functions, which are defined as ratios of the lengths of the sides of a right triangle.
The six trigonometric functions are: sine (sin), cosine (cos), tangent (tan) and cotangent (cot). Each of these functions is defined as the ratio of two sides of a right triangle.

###### Examples:

##### Solids

To understand what are solids, we prepared a quick revision for you:

Point

###### Point

is a location in space. It has neither shape nor size.

###### Straight Line

is a connected set of points that extends infinitely in two directions.

###### Planes

are a 2-dimensional surface made up of points that extends infinitely in all directions. There's only one plane that exists through noncollinear points.

- ###### The Relationship between Two Lines in Space:

  We distinguish three possible relationships that two lines can have.
  First is a pair of coplanar lines. These lines might intersect at any angle( as shown in diagram beneath) or they could be perpendicular( intersecting orthogonally).
  Second is example of pair of lines that do not intersect, will never meet. These lines are parallel.
  Last but not least we have two lines that neither intersect nor are parallel to one another. We call them skew lines. Skew lines can only exist in 3 dimensions.

- ###### The Relationship between a Line and Plane in Space

  There are three relationships that could occur between a line and a plane.
  A line might lie on a plane, resulting that every point will do so as well.
  A line that intersects a plane shares a common point that lies on both of them.
  Said line may intersact a plane orthogonally- which means that it wil be perpendicular to the plane. This results in line being perpendicular to all lines on that plane that intersect it.
  Third relationship between a line and plane happens when a line does not intersect a plane. When that occurs the line is parallel to the plane.

- ###### The Relationship between Two Planes in Space:

  Between two planes in space there are three possible relationships.
  If the two planes share all points, they are said to be coincident.
  If two planes intersect, the intersection is always a line. If said intersection occurs orthogonally, then it is perpendicular.
  When two planes do not intersect, they are parallel.

- ##### Prism

  Prism is a solid object bound on all its sides by plane faces. There are two types of faces in a prism. Top and bottom are identical and are called bases. We name prisms basing upon the shape of these bases( for example, a prism with traingualar base is called a triangular prism).

  Edge

  ###### Edge

  is a straight line that connects any two adjacent vertices of a prism.

  ###### Vertex

  a corner of a prism where any two edges meet.

  ###### Face

  is a closed, flat surface surrounded by vertices and edges.

  - ###### Examples of Prisms

    Square Prism

    ###### Square Prism

    ###### Cube

    ###### Triangular Prism

    ###### Pentagonal Prism

    ###### Irregular Pentagonal Prism

    The cross-section is not a "regular" shape making this prism an irregular one.

  - ###### Right and Oblique Prisms

    When the two ends are perfectly aligned it is a Right Prism otherwise it is an Oblique Prism:

  - ###### Mathematical Formulas

    The mathematical formulas for Surface Area, Volume and Diagonal of Square Prism:

    Surface Area

    ###### Surface Area

    Surface Area = 2 × Base Area + Base Perimeter × Length

    ###### Volume

    Volume = Base Area × Length

    ###### Diagonal of Square Prism

    Diagonal of Square Prism = √(a²+b²+c²)

- ##### Pyramids

  A pyramid is a three-dimensitonal figure with one flat polygon base. All the other faces are triangles and are called lateral faces. The number of lateral faces equals the number of sides on its base. Except for the base, all the faces connect at a vertex at the top that we call the apex.

  - ###### Examples of Pyramids

    Triangular Pyramid

    ###### Triangular Pyramid

    ###### Square Pyramid

    ###### Pentagonal Pyramid

    ###### Regular vs Irregular Pyramid

  - ###### Right and Oblique Pyramids

    When apex of pyramid is located exactly over the center of the base, then the pyramid is a right pyramid. In any other case it's oblique.

  - ###### Mathematical Formulas

    The mathematical formulas for Surface Area, Volume:

    Surface Area

    ###### Surface Area

    Surface Area = Base Area + (½ × perimeter of the base × Slant height)
    or
    Surface Area = Base Area + Lateral Area

    ###### Volume

    Volume = 1/3 × Base Area × Height

- ##### Cylinder

  A cylinder is a three-dimensitonal solid figure with two identical and parallel circular bases joined by a curved surface at a particular distance from the center (height of the cylinder). Cylinder has 3 faces, 2 edges and no vertices.

  - ###### Types of Cylinders

    Right Circular Cylinder

    ###### Right Circular Cylinder

    The axis of the two parallel bases is perpendicular to the center of the base.

    ###### Oblique Cylinder

    The axis of two parallel bases is not perpendicular to the center of the base.

    ###### Elliptic Cylinder

    The base of cylinder is in form of an elypse.

    ###### Right Circular Hollow Cylinder or Cylindrical Shell

    It's two right circular cylinders that are bounded one inside the other. The point of the axis is common and is perpendicular to the central base. However, it differs from the right circular cylinder since it is hollow in nature( there is some space or boid present inside).

  - ###### Mathematical Formulas

    The mathematical formulas for Surface Area, Curved Surface Area and Volume:

    Surface Area

    ###### Surface Area

    Total surface area of cylinder = 2πr(h+r)

    ###### Lateral Surface Are

    Surface Area = 2πr(h+r)

    ###### Volume

    Volume = πr²h

- ##### Cone

  A cone is a three-dimensional solid figure with a flat and curved surface pointed towards the top- a vertex. It has one circular face, zero edges and one vertex( corner).

  ###### Elements of Cone

  Radius of the Cone

  ###### Radius of the Cone

  is the distance between the center of teh circular base to any point on the rim of the base.

  ###### Height of the Cone

  is the distance between the corner of the cone to the center of the circular base.

  ###### Slant Height of the Cone

  is the distance between the corner of the cone and the point on the outer edge of circular base.

  - ###### Right and Oblique Cones

    There are two types of cones. A right one- where vertex is perpendicular to the center of base and an oblique one- where vertex isn't perpendicular to the center of the base.

  - ###### Mathematical Formulas

    The mathematical formulas for Surface Area, Curved Surface Area and Volume:

    Surface Area

    ###### Surface Area

    Total surface area of cone = πr(r+l)

    ###### Lateral Surface Area

    Surface Area = πrl

    ###### Volume

    Volume = (πr²h)/3

- ##### Sphere

  A sphere is a three-dimensional round-shaped object without any vertices or edges. All the points on surface are equally distant from its center.

  ###### Elements of Sphere

  Radius

  ###### Radius

  is the distance between the center of sphere and a point on sphere's surface.

  ###### Diameter

  is the distance between one point on the surface to the opposite point to it, passing through a center.

  ###### Circumference

  is the greatest cross-section of a circle that we can cut from a sphere.

  - ###### Mathematical Formulas

    The mathematical formulas for Surface Area,Volume, Circumference and Diameter:

    Surface Area

    ###### Surface Area

    4πr²

    ###### Volume

    Volume = (4πr³)/3

    ###### Circumference

    Volume = 2πr

    ###### Diameter

    Volume = 2r

- ##### Similar Solids

  Two solids are similar when distance between each point of one solid is proportional to distance between each point of second solid. In summary, similar solids have the same shape but not size.

  ###### Basic understanding

  Scale Factor Ratio

  ###### Scale Factor Ratio

  Distance between each of points in one solid must be proportional to distance between each of points in other solid.
  a/b

  ###### Surface Area Ratio

  If two solids are similat with a scale factor a/b then the surface areas are in a ratio of (a/b)²

  ###### Volume Ratio

  If two solids are similat with a scale factor a/b then the volumes of similar solids are in a ratio of (a/b)³