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Geometry

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Geometry

Geometry examines the sizes, shapes and distances of objects and compares the relationships between points, lines, curves, angles, surfaces and solids.

The term Geometry stems from the ancient Greek word, geōmetría, which translates to land measurement in English. It happens to be one of the oldest branches of Mathematics that examines spatial qualities relating to distance, shape, size, figures and relative position. Some historians have noted that the origins of Geometry date back to the 2nd millennium BC in ancient Mesopotamia and Egypt.

Geometry is a branch of mathematics that studies the properties of figures in space.

This topic is divided into two segments: Plane Geometry and Solid Geometry.

  1. Plane Geometry: The study of flat surfaces in two-dimensional space
  2. Solid Geometry: The study of objects in three-dimensional space
    • Example: cubes, cylinders, spheres

In this topic, we shall observe the contents that comprise Plane Geometry and Solid Geometry to give us a better understanding of Geometry. For simplicity, these two segments will be further divided into smaller sections.

Plane Geometry

The idea of Plane Geometry is derived from looking at objects from a two-dimensional perspective. Let us take a look!

There are three concepts to consider in Plane Geometry.

  1. A point: this represents the position and has no dimensions (yellow points A, B and C).

  2. A line: this is a straight segment in one dimension, with no beginning and no end (red line).

  3. A plane: this is a flat surface that extends indefinitely in two dimensions (blue plane).

Point, line and plane, Aishah Amri - StudySmarter Originals

These concepts constitute the basic objects from which all Geometry can be constructed, in other words, any other geometrical object can be defined in terms of a combination of these three concepts.

Cartesian Coordinate System

Imagine we would like to study the properties of the triangle below. We would like to know the measure of its angles and sides as well as how much space it occupies (this is also known as the area). However, we were not given any suitable measuring instrument such as a ruler to determine this. How do we go about this problem then? Here is where the Cartesian Coordinate System enters the scene.

Example 1, Aishah Amri - StudySmarter Originals

The system of Cartesian coordinates was created by a French Mathematician and Philosopher named René Descartes and was developed as a way to represent a plane. Coincidentally, his name in Latin is Cartesius, thence "Cartesian". Isn't that neat?

This system makes it easier to represent and locate points, lines and any other shape in a plane. It is such a powerful and simple system that has completely changed the way we work Geometry mathematically!

A figure illustrated in a Cartesian coordinate system in two dimensions represented by the x-axis and y-axis. A point is denoted by the coordinates (x, y) on the Cartesian plane.

The x-value in the point (x, y) is called the abscissa.

The y-value in the point (x, y) is called the ordinate.

The two dimensions here refer to the length and height of the figure. The point of intersection between the x-axis and the y-axis is called the origin and is denoted by the letter O. The coordinates of the origin is (0, 0).

The cartesian coordinate system contains four quadrants, listed below.

  1. Quadrant I: Refers to a point located in both positive regions of the x-axis and y-axis.

  2. Quadrant II: Refers to a point located in the negative region of the x-axis and the positive region of the y-axis.

  3. Quadrant III: Refers to a point located in both negative regions of the x-axis and y-axis.

  4. Quadrant IV: Refers to a point located in the positive region of the x-axis and the negative region of the y-axis.

Below is a graphical representation of the Cartesian coordinate system.

Quadrant system, Aishah Amri - StudySmarter Originals

Let us return to our triangle, introduced at the beginning of this section. With these concepts in place, let us position this triangle on the Cartesian plane.

Example 2, Aishah Amri - StudySmarter Originals

Here, our triangle is represented by 3 points A, B and C and 3 line segments AB, AC and BC. With this information, we can definitely calculate the required measures for this triangle. Though more on this later For now, let us stick to our Quadrant system with the following example.

The point (3, 2) is located in the first quadrant in the Cartesian coordinate system.

The point (2, –1) is located in the fourth quadrant in the Cartesian coordinate system. This is illustrated below.

Example 3, Aishah Amri - StudySmarter Originals

Lines

A line is represented by the equation y = mx + c, where m is the slope or gradient of the line and c is the y-intercept.

The gradient measures the steepness of a line and is given by the formula:

.

Two lines are said to be parallel if they lie on the same plane and do not intersect each other. A pair of parallel lines have the same slope.

Parallel lines, Aishah Amri - StudySmarter Originals

It is important to note that a pair of parallel lines do not intersect each other no matter how far you extend them.

Two lines are said to be perpendicular if they intersect each other at right angles. The product of the two slopes is –1.

Perpendicular lines, Aishah Amri - StudySmarter Originals

A line segment is a line with two endpoints.

Line segment, Aishah Amri - StudySmarter Originals

A ray is a line with a fixed starting point an endpoint that goes on forever.

Ray, Aishah Amri - StudySmarter Originals

Midpoint and Distance Formula

Midpoint Formula locates the point that is equidistant from two endpoints on a line segment.

The Distance Formula calculates the length between two points on a line.

Angles

Angles are useful when it comes to describing polygons such as quadrilaterals and triangles as we shall see later on in this lesson. Let us first define an angle.

An angle is formed by the union of two rays. These rays meet at a common endpoint. It is represented by the symbol ∠.

Below are several notable types of angles you should familiarize yourself with.

Types of Angles
Diagram
Description
Acute Angle

Acute angle, Aishah Amri - StudySmarter Originals

Less than 90o
Right Angle

Right angle, Aishah Amri - StudySmarter Originals

Equal to 90o
Obtuse Angle

Obtuse angle, Aishah Amri - StudySmarter Originals

More than 90o
Straight Angle

Straight angle, Aishah Amri - StudySmarter Originals

Equal to 180o
Reflex Angle

Reflex angle, Aishah Amri - StudySmarter Originals

More than 180o
Full Rotation

Full rotation, Aishah Amri - StudySmarter Originals

Equal to 360o

An interior angle is an angle inside a shape and is formed by two sides of the polygon.

An exterior angle is an angle between any side of a shape and a line extended from the next side of the polygon.

Two angles are called supplementary if they add up to 180o.

Two angles are said to be complementary if they add up to 90o.

Vectors

A vector is a concept that is important when it comes to describing movement from one point to another.

A vector is an object that has both magnitude and direction.

By the definition above, a vector quantity has both direction and magnitude (size). A vector can be visualised geometrically as a directed line segment with a length equal to the magnitude of the vector and a direction indicated by an arrow. Below is a graphical representation of a vector.

Vector, Aishah Amri - StudySmarter Originals

Let us now look at some common vector operations in the table below.

Vector Operations
Formula
Graphical Representation
Addition

Vector addition, Aishah Amri - StudySmarter Originals

Subtraction

Vector subtraction, Aishah Amri - StudySmarter Originals

Scalar Product

Scalar product, Aishah Amri - StudySmarter Originals

Dot Product

or

Dot product, Aishah Amri - StudySmarter Originals

Perimeter and Area

The perimeter is the distance around the edges of an object.

The area of an object is the size of its surface.

Find the perimeter and area of the rectangle below.

Example 4, Aishah Amri - StudySmarter Originals

Solution

The perimeter of a rectangle is the sum of all its sides. Thus,

P = 2 + 2 + 3 + 3 = 10 units

The area of a rectangle is found by multiplying its length and width together. In doing so, we obtain

A = 2 x 3 = 6 units2

Therefore, the perimeter of the rectangle is 10 units and its area is 6 units2

Congruence and Similarity

Congruence and similarity serve an important role in Geometry when it comes to comparing shapes and finding the measures between them.

Two objects are said to be congruent if they are of equal shape and size.

Two objects are said to be similar if they have the same shape but not the same size.

The triangles below are congruent as the lengths of their sides are the same.

Example 5, Aishah Amri - StudySmarter Originals

The squares below are similar as they are of the same shape but the lengths of their sides are different.

Example 6, Aishah Amri - StudySmarter Originals

Transformations

In this section, we shall become acquainted with the concept of transformations. Transformations help us visualize objects on a plane in different orientations.

In geometry, a transformation is a term used to describe a change towards a given shape.

Type of Transformation
Description
Example
Rotation
Turning an object about its centre

Rotation, Aishah Amri - StudySmarter Originals

Reflection
Flipping and object about a line

Reflection, Aishah Amri - StudySmarter Originals

Translation
Shifting an object given a direction

Translation, Aishah Amri - StudySmarter Originals

Dilation
Resizing an object given a magnitude

Dilation, Aishah Amri - StudySmarter Originals

Symmetry

Symmetry is an important concept when it comes to reproducing shapes without changing their original form. Let us dive into its definition and become familiar with three primary types of symmetry as described in the table below.

The term symmetry refers to a shape that maintains its form when it is moved, rotated, or flipped. An object is said to be symmetrical if it contains two matching halves.

Type of Symmetry
Description
Example
Reflection Symmetry
A form of symmetry that mirrors an object

Reflection symmetry, Aishah Amri - StudySmarter Originals

Rotational Symmetry
A property in which a shape looks the same after a rotation or partial turn

Rotational symmetry, Aishah Amri - StudySmarter Originals

Point Symmetry
Two same objects are reflected in opposite directions and are equidistant from a central point

Point symmetry, Aishah Amri - StudySmarter Originals

Polygons

Previously, we have defined lines, points and planes. Now, what if we joined several lines together at their endpoints on a plane. What do we get from this construction? This, in fact, would result in a polygon!

A polygon is a two-dimensional shape made up of straight lines.

If all the sides and all the angles of a polygon are equal, it is called a regular polygon. Otherwise, it is called an irregular polygon.

Property
Description
Exterior Angle of a Polygon
The sum of the exterior angles of a polygon is 360o
For a polygon with n sides, each exterior angle is equal to
Exterior Angle =
Interior Angle of a Polygon
For a polygon with n sides, each interior angle of a polygon is given by the formula
Interior Angle = 180o - Exterior Angle

A diagonal is a line segment from one corner to another corner of a polygon.

Diagonal, Aishah Amri - StudySmarter Originals

A point at which two diagonals meet is called a point of intersection.

These lines are not parallel to one another and the slopes are reciprocals of each other.

Triangles

Triangles, as you shall see throughout Geometry, play an important role in another subtopic called Trigonometry. Though, more on that later! Here, we shall only cover the area of a basic triangle and describe the six main types of triangles we shall commonly see throughout this syllabus.

A triangle is a polygon with three sides and three vertices. The sum of the interior angles of a triangle is 180o.

The area of a triangle is given by the formula

,

where b is the base and h is the height.

Area of a triangle, Aishah Amri - StudySmarter Originals

Type of Triangle
Properties
Diagram
Equilateral Triangle
Three equal sides and three equal angles

Equilateral triangle, Aishah Amri - StudySmarter Originals

Isosceles Triangle
Two equal sides and two equal angles

Isosceles triangle, Aishah Amri - StudySmarter Originals

Scalene Triangle
No equal sides and no equal angles

Scalene triangle, Aishah Amri - StudySmarter Originals

Acute Triangle
All angles are less than 90o

Acute triangle, Aishah Amri - StudySmarter Originals

Right Triangle
Has one angle equal to 90o

Right triangle, Aishah Amri - StudySmarter Originals

Obtuse Triangle
Has one angle more than 90o

Obtuse triangle, Aishah Amri - StudySmarter Originals

Quadrilaterals

Next, we shall look at another form of polygons called quadrilaterals. The table below describes several types of quadrilaterals along with their properties and area formula.

A quadrilateral is a polygon with four sides (edges) and four vertices (corners). The sum of the interior angles of a quadrilateral is 360o.

Type of Quadrilateral
Diagram
Properties
Area

Rectangle, Aishah Amri - StudySmarter Originals

Opposite sides are equal

4 right angles

Opposite sides are parallel

lh
Square

Square, Aishah Amri - StudySmarter Originals

4 equal sides

4 right angles

Opposite sides are parallel

l2
Trapezoid

Trapezoid, Aishah Amri - StudySmarter Originals

2 parallel sides

Parallelogram

Paralellogram, Aishah Amri - StudySmarter Originals

Opposite sides are equal

Opposite sides are parallel

bh
Rhombus

Rhombus, Aishah Amri - StudySmarter Originals

4 right angles

Opposite sides are parallel

Circles

Let us move on to another shape of interest called circles. Here, we shall also discover the components that make up a circle.

A circle is a set of points that are equidistant from a point, called the centre.

Concept
Diagram
Description
Components of a Circle

Components of a Circle, Aishah Amri - StudySmarter Originals

  • The radius is the distance from the centre to the edge of the circle
  • The diameter is the straight length across the circle, through the centre
  • The circumference is the distance around the edge of the circle
Circumference and Area of a Circle

Circumference and Area of a Circle, Aishah Amri - StudySmarter Originals

d = 2r
C = 2πr = dπ
A = πr2
where d = diameter, r = radius, C = circumference and A = area
Lines on a Circle

Lines on a Circle, Aishah Amri - StudySmarter Originals

  • A tangent is a line that touches a point on the edge of a circle
  • A secant is a line that cuts the circle at two points
  • A chord is a line segment that passes one point to another point on the circumference of a circle
  • The arc is a part of the circumference of a circle
Sector of a Circle

Sector of a Circle, Aishah Amri - StudySmarter Originals

The sector refers to a 'slice' of the circle. The area of a sector is given by the formula
Segment of a Circle

Segment of a Circle, Aishah Amri - StudySmarter Originals

A segment is a part that is cut from the circle by a chord. The area of a segment is given by the formula
Arc Length of a Circle

Arc length of a Circle, Aishah Amri - StudySmarter Originals

The arc length of a sector (or segment) of a circle is given by the formula
Annulus

Annulus, Aishah Amri - StudySmarter Originals

An annulus is made up of two circles with the same centre. The radius of these two circles is different. The shape of an annulus resembles a ring. The area of the blue region is given by the formula

Solid Geometry

Let us now move on to the next vital section of this topic called Solid Geometry. Here, we shall visualize objects in three-dimensional space.

A solid is called three-dimensional as it is described by an object in three dimensions.

These dimensions are called the width (sometimes referred to as the base), length and height of an object.

There are two types of solids to consider in this section.

  1. Polyhedra: Any solid with only flat faces
    • Example: Cubes, Pyramids, Prisms
  2. Non-Polyhedra: Any solid with at least one curved face
    • Example: Spheres, Cylinders, Cones

A solid is often illustrated in a Cartesian coordinate system in three dimensions represented by the x-axis, y-axis and z-axis. Below is a graphical representation of a sphere centred at the origin with a radius of 2 units. The red line represents the x-axis, the green line denotes the y-axis and the blue line defines the z-axis.

Three-dimensional Cartesian coordinate system of a sphere, Aishah Amri - StudySmarter Originals

Properties of Solids

All solids have two characteristics that define their form.

  1. Surface area

  2. Volume

Another way to distinguish different solids from each other is by observing the number of vertices, edges and faces they own.

Faces, Edges and Vertices

We shall first describe what these components mean for a solid and present a table illustrating several solids along with their number of faces, edges and vertices.

The face refers to a flat surface on a solid.

The curved face describes a curved surface.

An edge is a line segment in which two faces meet.

A vertex (or corner) is a point in which two edges meet.

Solid
Diagram
Number of Faces
Number of Edges
Number of Vertices
Number of Curved Faces
Sphere

Sphere, Aishah Amri - StudySmarter Originals

0
0
0
1
Ellipsoid

Ellipsoid, Aishah Amri - StudySmarter Originals

0
0
0
1
Cone

Cone, Aishah Amri - StudySmarter Originals

1
1
1
1
Cylinder

Cylinder, Aishah Amri - StudySmarter Originals

2
2
0
1
Tetrahedron

Tetrahedron, Aishah Amri - StudySmarter Originals

4
6
4
0
Square Pyramid

Square pyramid, Aishah Amri - StudySmarter Originals

5
8
5
0
Triangular Prism

Triangular prism, Aishah Amri - StudySmarter Originals

5
9
6
0
Cube

Cube, Aishah Amri - StudySmarter Originals

6
12
8
0
Cuboid

Cuboid, Aishah Amri - StudySmarter Originals

6
12
8
0
Octahedron

Octahedron, Aishah Amri - StudySmarter Originals

8
12
6
0
Pentagonal Prism

Pentagonal prism, Aishah Amri - StudySmarter Originals

7
15
10
0
Hexagonal Prism

Sphere, Aishah Amri - StudySmarter Originals

8
18
12
0

Surface Area and Volume

In this section, we shall exhibit a table that describes the formula of the surface area and volume of a few notable solids.

Solid
Diagram
Surface Area
Volume
Notation
Sphere

Sphere, Aishah Amri - StudySmarter Originals

4πr2
r = radius
Hemisphere

Hemisphere, Aishah Amri - StudySmarter Originals

3πr2
r = radius
Cone

Cone, Aishah Amri - StudySmarter Originals

πr (s + r)
r = radius
s = slant height
h = height
Cylinder

Cylinder, Aishah Amri - StudySmarter Originals

2πr (r + h)
πr2h
r = radius
h = height
Pyramid

Pyramid, Aishah Amri - StudySmarter Originals

bl + 2bs
l = length
b = base
h = height
s = slant height
Cube

Cube, Aishah Amri - StudySmarter Originals

6l2
l3
l = length
Cuboid

Cuboid, Aishah Amri - StudySmarter Originals

2 (lb + bh + lh)
lbh
l = length
b = base
h = height
Triangular Prism

Triangular prism, Aishah Amri - StudySmarter Originals

bh + lb + 2ls
l = length
b = base
h = height
s = slant height
Trapezoidal Prism

Trapezoidal prism, Aishah Amri - StudySmarter Originals

(a + b)h + bl + al + 2ls
l = length
b = base
h = height
s = slant height
a = top length

Cross-Section of a Solid

Another important concept that falls under the category of Solid Geometry is called the cross-section.

A cross-section is a shape made by cutting through a solid with a plane.

The cross-section of a cylinder cut by a horizontal plane gives us a circle.

Example 7, Aishah Amri - StudySmarter Originals

Euler's Formula

Euler's Formula states that for any polyhedron that does not intersect itself or have any holes, the number of faces plus the number of vertices minus the number of edges always equals two. This can be written by the expression below.

Let us look at an example that applies this formula.

Verify that Euler's Formula is satisfied for a square pyramid.

Solution

From our table above, a pyramid has the following features:

Number of Faces: 5

Number of Vertices: 5

Number of Edges: 8

Now, applying Euler's Formula, we obtain

Thus, Euler's Formula holds true for a square pyramid.

Geometry - Key takeaways

  • Geometry studies the properties of figures in space.
  • Geometry is branched into two parts:
    1. Plane Geometry - studies flat surfaces in two-dimensional space
    2. Solid Geometry - studies objects in three-dimensional space
  • Important Concepts in Plane Geometry
    Concept
    Explanation
    Point, line and plane

    Point: represents the position and has no dimensions

    Line: a straight segment in one dimension

    Plane: a flat surface that extends indefinitely in two dimensions

    Angles

    Interior angle: angle inside a shape, formed by two sides of the polygon

    Exterior angle: angle between any side of a shape and a line extended from the next side of the polygon

    Vectors

    Describes the direction and magnitude of an object

    Perimeter and Area

    Perimeter: the distance around the edges of an object

    Area: the size of its surface

    Congruency and Similarity

    Two objects are congruent if they are of equal shape and size

    Two objects are similar if they have the same shape but not the same size

    Types of Transformations
    Rotation, reflection, translation, dilation
    Types of Symmetry
    Reflection symmetry, rotational symmetry, point symmetry
    A two-dimensional shape made up of straight lines
    Triangles: a polygon with three sides and three vertices
    Quadrilaterals: a polygon with four sides and four vertices
    Circles
    Set of points that are equidistant from a centre
  • Important Concepts in Solid Geometry
    Concept
    Explanation
    Types
    Polyhedra: a solid with only flat faces
    Non-polyhedra: a solid with at least one curved face
    Properties
    Has volume and surface area
    Components
    Face: flat surface on a solid
    Edge: line segment in which two faces meet
    Vertex: point in which two edges meet
    Cross-section

    A shape made by cutting through a solid with a plane

    Euler's Formula
    F + V - E = 2

Frequently Asked Questions about Geometry

Geometry is a branch of mathematics that studies the sizes, shapes, positions, angles and dimensions of a particular object. 

A converse statement is an argument constructed by reversing the hypothesis and the conclusion.  

In geometry, two objects are congruent if they are exactly the same shape and size.

The geometry triangle rules are the Sine, Cosine and Tangent Rules.

The area of a triangle is the product of the height and base of a triangle, multiplied by half.

Final Geometry Quiz

Question

What is geometry?

Show answer

Answer

Geometry is a branch of mathematics that studies the properties of figures in space

Show question

Question

What are the two main components of geometry?

Show answer

Answer

  1. Plane geometry
  2. Solid geometry

Show question

Question

What is plane geometry?

Show answer

Answer

Plane geometry studies flat surfaces in two-dimensional space 

Show question

Question

What is solid geometry?

Show answer

Answer

Solid Geometry studies objects in three-dimensional space

Show question

Question

What is the difference between a point and a line?

Show answer

Answer

A point represents the position and has no dimensions while a line is a straight segment in one dimension

Show question

Question

What is the difference between a point and a plane?

Show answer

Answer

A point represents the position and has no dimensions and a plane is a flat surface that extends indefinitely in two dimensions

Show question

Question

What is an interior angle?

Show answer

Answer

An angle inside a shape, formed by two sides of the polygon 

Show question

Question

What is an exterior angle?

Show answer

Answer

An angle between any side of a shape and a line extended from the next side of the polygon 

Show question

Question

What describes a vector?

Show answer

Answer

A vector describes the direction and magnitude of an object 

Show question

Question

What is the difference between a perimeter and the area of a 2D object?

Show answer

Answer

The perimeter is the distance around the edges of an object while the area is the size of its surface

Show question

Question

What does congruency mean in geometry?

Show answer

Answer

Two objects are congruent if they are of equal shape and size

Show question

Question

What does similarity mean in geometry?

Show answer

Answer

Two objects are similar if they have the same shape but not the same size

Show question

Question

What is a polygon?

Show answer

Answer

A polygon is a two-dimensional shape made up of straight lines 

Show question

Question

What defines a triangle?

Show answer

Answer

A polygon with three sides and three vertices whose interior angles add up to 180o

Show question

Question

What defines a quadrilateral?

Show answer

Answer

A polygon with four sides and four vertices whose interior angles add up to 360o

Show question

Question

What defines a circle?

Show answer

Answer

A set of points that are equidistant from a central point called the centre 

Show question

Question

Name the two types of solids

Show answer

Answer

  1. Polyhedra
  2. Non-polyhedra

Show question

Question

What is the difference between a polyhedron and a non-polyhedron?

Show answer

Answer

A polyhedra is a solid with only flat faces while a non-polyhedra is a solid with at least one curved face 

Show question

Question

What are the two main properties of all solids?

Show answer

Answer

All solids have volume and surface area

Show question

Question

Define the face, vertex and edge of  solid

Show answer

Answer

  1. The face is a flat surface on a solid
  2. The edge is a line segment in which two faces meet
  3. The vertex is a point in which two edges meet 

Show question

Question

What is a cross-section?

Show answer

Answer

A cross-section is a shape made by cutting through a solid with a plane 

Show question

Question

In the context of geometry, what does Euler's Formula tell us about a solid?

Show answer

Answer

The number of face plus the number of vertices minus the number of edges always equal to two

Show question

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