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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.
- Plane Geometry: The study of flat surfaces in two-dimensional space
- Example: lines, circles, polygons
- 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.
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.
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.
Quadrant I: Refers to a point located in both positive regions of the x-axis and y-axis.
Quadrant II: Refers to a point located in the negative region of the x-axis and the positive region of the y-axis.
Quadrant III: Refers to a point located in both negative regions of the x-axis and y-axis.
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.
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 360oFor a polygon with n sides, each exterior angle is equal toExterior Angle = |
| Interior Angle of a Polygon | For a polygon with n sides, each interior angle of a polygon is given by the formulaInterior 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 |
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 |
|
| Circumference and Area of a Circle |  Circumference and Area of a Circle, Aishah Amri - StudySmarter Originals | d = 2rC = 2πr = dπA = πr2where 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 |
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.
- Polyhedra: Any solid with only flat faces
- Example: Cubes, Pyramids, Prisms
- 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.
Surface area
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 = radiuss = slant heighth = height | |
| Cylinder | Cylinder, Aishah Amri - StudySmarter Originals | 2πr (r + h) | πr2h | r = radiush = height |
| Pyramid |
Pyramid, Aishah Amri - StudySmarter Originals | bl + 2bs | l = lengthb = baseh = heights = slant height | |
| Cube |
Cube, Aishah Amri - StudySmarter Originals | 6l2 | l3 | l = length |
| Cuboid |
Cuboid, Aishah Amri - StudySmarter Originals | 2 (lb + bh + lh) | lbh | l = lengthb = baseh = height |
| Triangular Prism |
Triangular prism, Aishah Amri - StudySmarter Originals | bh + lb + 2ls | l = lengthb = baseh = heights = slant height | |
| Trapezoidal Prism |
Trapezoidal prism, Aishah Amri - StudySmarter Originals | (a + b)h + bl + al + 2ls | l = lengthb = baseh = heights = slant heighta = 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:
- Plane Geometry - studies flat surfaces in two-dimensional space
- 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 Polygons A two-dimensional shape made up of straight linesTriangles: a polygon with three sides and three verticesQuadrilaterals: 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 facesNon-polyhedra: a solid with at least one curved face Properties Has volume and surface area Components Face: flat surface on a solidEdge: line segment in which two faces meetVertex: 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
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Frequently Asked Questions about Geometry
What is area of triangle in geometry?
The area of a triangle is the product of the height and base of a triangle, multiplied by half.
What is geometry in maths?
Geometry is a branch of mathematics that studies the sizes, shapes, positions, angles and dimensions of a particular object.
What is a converse in geometry?
A converse statement is an argument constructed by reversing the hypothesis and the conclusion.
What does congruent mean in geometry?
In geometry, two objects are congruent if they are exactly the same shape and size.
What are geometry triangle rules?
The geometry triangle rules are the Sine, Cosine and Tangent Rules.
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