StudySmarter  The allinone study app.
4.8 • +11k Ratings
More than 3 Million Downloads
Free
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.
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.
The idea of Plane Geometry is derived from looking at objects from a twodimensional perspective. Let us take a look!
There are three concepts to consider in Plane Geometry.
A point: this represents the position and has no dimensions (yellow points A, B and C).
A line: this is a straight segment in one dimension, with no beginning and no end (red line).
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.
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 xaxis and yaxis. A point is denoted by the coordinates (x, y) on the Cartesian plane.
The xvalue in the point (x, y) is called the abscissa.
The yvalue 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 xaxis and the yaxis 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.
Quadrant I: Refers to a point located in both positive regions of the xaxis and yaxis.
Quadrant II: Refers to a point located in the negative region of the xaxis and the positive region of the yaxis.
Quadrant III: Refers to a point located in both negative regions of the xaxis and yaxis.
Quadrant IV: Refers to a point located in the positive region of the xaxis and the negative region of the yaxis.
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
A line is represented by the equation y = mx + c, where m is the slope or gradient of the line and c is the yintercept.
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 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 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 90^{o} 
Right Angle  Right angle, Aishah Amri  StudySmarter Originals  Equal to 90^{o} 
Obtuse Angle  Obtuse angle, Aishah Amri  StudySmarter Originals  More than 90^{o} 
Straight Angle  Straight angle, Aishah Amri  StudySmarter Originals  Equal to 180^{o} 
Reflex Angle  Reflex angle, Aishah Amri  StudySmarter Originals  More than 180^{o} 
Full Rotation  Full rotation, Aishah Amri  StudySmarter Originals  Equal to 360^{o} 
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 180^{o}.
Two angles are said to be complementary if they add up to 90^{o}.
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 
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 units^{2}
Therefore, the perimeter of the rectangle is 10 units and its area is 6 units^{2}
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
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 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 
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 twodimensional 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 360^{o} 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 = 180^{o}  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, 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 180^{o}.
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 90^{o}  Acute triangle, Aishah Amri  StudySmarter Originals 
Right Triangle  Has one angle equal to 90^{o}  Right triangle, Aishah Amri  StudySmarter Originals 
Obtuse Triangle  Has one angle more than 90^{o}  Obtuse triangle, Aishah Amri  StudySmarter Originals 
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 360^{o}.
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  l^{2} 
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 
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 

Circumference and Area of a Circle  Circumference and Area of a Circle, Aishah Amri  StudySmarter Originals  d = 2r C = 2πr = dπ A = πr^{2} where d = diameter, r = radius, C = circumference and A = area 
Lines on a Circle  Lines on a Circle, Aishah Amri  StudySmarter Originals 

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 
Let us now move on to the next vital section of this topic called Solid Geometry. Here, we shall visualize objects in threedimensional space.
A solid is called threedimensional 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.
A solid is often illustrated in a Cartesian coordinate system in three dimensions represented by the xaxis, yaxis and zaxis. Below is a graphical representation of a sphere centred at the origin with a radius of 2 units. The red line represents the xaxis, the green line denotes the yaxis and the blue line defines the zaxis.
Threedimensional Cartesian coordinate system of a sphere, Aishah Amri  StudySmarter Originals
All solids have two characteristics that define their form.
Surface area
Volume
Another way to distinguish different solids from each other is by observing the number of vertices, edges and faces they own.
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 
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  4πr^{2}  r = radius  
Hemisphere  Hemisphere, Aishah Amri  StudySmarter Originals  3πr^{2}  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)  πr^{2}h  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  6l^{2}  l^{3}  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 
Another important concept that falls under the category of Solid Geometry is called the crosssection.
A crosssection is a shape made by cutting through a solid with a plane.
The crosssection of a cylinder cut by a horizontal plane gives us a circle.
Example 7, Aishah Amri  StudySmarter Originals
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.
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 twodimensional 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 
Concept  Explanation 
Types  Polyhedra: a solid with only flat faces Nonpolyhedra: 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 
Crosssection  A shape made by cutting through a solid with a plane 
Euler's Formula  F + V  E = 2 
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.
Be perfectly prepared on time with an individual plan.
Test your knowledge with gamified quizzes.
Create and find flashcards in record time.
Create beautiful notes faster than ever before.
Have all your study materials in one place.
Upload unlimited documents and save them online.
Identify your study strength and weaknesses.
Set individual study goals and earn points reaching them.
Stop procrastinating with our study reminders.
Earn points, unlock badges and level up while studying.
Create flashcards in notes completely automatically.
Create the most beautiful study materials using our templates.
Sign up to highlight and take notes. It’s 100% free.