Geometry and trigonometry
Geometry is the branch of mathematics that looks at the properties of shapes, angles and lines. This page gives an introduction to the topic. Later pages will give more detail on properties of triangles, how to calculate perimeter and area of two-dimensional shapes, and surface area, volume and density of three-dimensional shapes.
In brief, this page covers:
- Different types of angles
- Different types of polygons
- Key properties of circles, including how their circumference, diameter and radius are related
- Different types of three-dimensional shapes
- How to work out whether a shape has one or more lines of symmetry
Angles
An angle is the amount of rotation between two lines that meet (or intersect) at a point. For example, in the diagram below, there is an angle between the lines \(AB\) and \(AC\). We can call this angle either \(\angle CAB\) or \(\angle A\).
Angles are usually measured in degrees, shown by the symbol \(^\circ\). One full turn, or rotation, is equal to \(360^\circ.\). Angles can also be measured in radians, where \(\pi\,\textrm{rad} = 180^\circ\) and \(2\pi\,\textrm{rad} = 360^\circ,\). However, this module will only use degrees.
There are names for different angles depending on their size, with some of the most common ones to be aware of as follows:
Acute angles are angles that are less than \(90^\circ\)
Right angles are angles that are equal to \(90^\circ\). This is usually represented with a small square in the angle.
Obtuse angles are angles that are bigger than \(90^\circ\) but less than \(180^\circ\)
\(\)
Angles can also be grouped according to what they add up to when combined with another angle. Two of the most common examples of this are:
Complementary angles are two angles that add up to \(90^\circ\)
If you know one angle, you can find its complimentary angle by subtracting it from \(90^\circ\)
Supplementary angles are two angles that add up to \(180^\circ\)
If you know one angle, you can find its complimentary angle by subtracting it from \(180^\circ\)
\(\)
Check your understanding of different types of angles by completing one or both of the following activities:
The supplementary angle of \(115^\circ\) is \(65^\circ\) since \(180^\circ - 115^\circ = 65^\circ\)
The complementary angle of \(42^\circ\) is \(48^\circ\) since \(90^\circ - 42^\circ = 48^\circ\)
Polygons
A polygon is a two-dimensional shape made up of straight sides. When all the sides of a polygon are the same length and all the angles are equal, it is called a regular polygon. The table below lists details of some common types of polygons:
| Polygon name | Number of sides | Number of angles | Sum of angles |
|---|---|---|---|
| Triangle | \(3\) | \(3\) | \(180^\circ\) |
| Quadrilateral | \(4\) | \(4\) | \(360^\circ\) |
| Pentagon | \(5\) | \(5\) | \(540^\circ\) |
| Hexagon | \(6\) | \(6\) | \(720^\circ\) |
| Heptagon | \(7\) | \(7\) | \(900^\circ\) |
| Octagon | \(8\) | \(8\) | \(1080^\circ\) |
| Nonagon | \(9\) | \(9\) | \(1260^\circ\) |
| Decagon | \(10\) | \(10\) | \(1440^\circ\) |
You can work out the sum of the internal angles of a polygon, as shown in the table, by splitting the polygon into triangles. Then multiply the number of triangles by \(180^\circ\) (since this is the sum of the angles in a triangle). For example, a pentagon can be split into three triangles, as shown below. This gives \(3 \times 180^\circ = 540^\circ\)
Alternatively, you can use the formula \((2n - 4) \times 90^\circ\) to work out the sum of the angles, where \(n\) is the number of sides of the polygon. For example, using the formula for a pentagon gives the same result as before:
\[\begin{aligned} (2 \times 5 - 4) \times 90^\circ &= (10 - 4) \times 90^\circ \\\ &= 6 \times 90^\circ \\\ &= 540^\circ \\\ \end{aligned}\]Check your knowledge of different types of polygons by completing one or both of the following activities:
\(1080^\circ \div 180^\circ = 6\) triangles (or you could draw a diagram to calculate)
\((2 \times 12 - 4) \times 90^\circ = 20 \times 90^\circ = 1800^\circ\) (or you could draw a diagram to calculate)
For details of how to calculate the perimeter and area of some of the polygons listed, see the Perimeter and area page of this module. You can also view more information about triangles on the Trigonometry page.
Circles
A circle is a shape made by a curved line that is always the same distance from a central point. This distance, from the centre of the circle to the outside edge, is called the radius. The diameter is the distance from one side of the circle to the other, passing through the centre. This means the diameter is double the radius. The circumference is the distance around the outside of the circle.
No matter what size a circle is, the ratio of the circumference to the diameter is always the same. This ratio is an irrational number called pi (written as \(\pi\)), which is approximately \(3.14\). Irrational means that the decimal places in pi go on forever without repeating:
\[\pi = \frac{circumference}{diameter}\]Because of this, you can calculate the circumference of a circle if you know the the diameter or the radius, and vice versa. The formulas below show how, using \(c\) to represent the circumference, \(d\) the diameter and \(r\) the radius:
\(c = \pi d\)
\(c = 2 \pi r\)
A sector is the ‘slice’ of a circle formed by two radii and an arc of the circumference. If the central angle of the sector (the angle formed by the two radii) is less than \(180^\circ\), it is called a minor sector. If it is equal to \(180^\circ\), it is called a semicircle. If it is more than \(180^\circ\), it is called a major sector.
\(\) Check your understanding of the properties of circles by completing one or both of the following activities:
For details of how to calculate the area of a circle or the perimeter or area of a sector of a circle, see the Perimeter and area page of this module.
Three-dimensional shapes
Some common examples of three-dimensional shapes are cubes, prisms, pyramids, cylinders, spheres and cones. These shapes are explained below. Information on how to calculate the surface area and volume of each is in the Surface area, volume and density page of this module.
Cubes
A cube is a three-dimensional shape with six identical square faces. Each face is perpendicular to the faces next to it. A cube is a special type of prism.
Prisms
A prism is a three-dimensional shape with two identical polygon bases (a top and bottom face) and sides made of squares, rectangles or parallelograms. If the bases of a prism are regular polygons, the prism is called a regular prism. It is usually named after the shape of its base. For example, a prism with triangular bases is called a triangular prism, and a prism with rectangular bases is called a rectangular prism (also known as a cuboid). If one of the bases is missing, the prism is called an open prism rather than a closed prism.
Pyramids
A pyramid is a three-dimensional shape with a polygon base and triangular sides that meet at a point at the top, called the apex. If the base of a pyramid is a regular polygon, the pyramid is called a regular pyramid. It is usually named after the shape of its base. For example, a pyramid with a triangular base is called a triangular pyramid (also known as a tetrahedron, or a regular tetrahedron if all its faces are equilateral triangles). A pyramid with a square base is called a square pyramid.
Cylinders
A cylinder is a three-dimensional shape with two identical flat circular ends joined by a curved side. If one of the circular ends is missing, the cylinder is called an open cylinder rather than a closed cylinder.
Spheres
A sphere is a three-dimensional shape with a curved surface that is always the same distance from a central point. This distance, from the centre of the sphere to the outside, is called the radius. The diameter is the distance from one side of the sphere to the other, passing through the centre. This means the diameter is double the radius.
Cones
A cone is a three-dimensional shape with a circular base and a curved surface that narrows to a point called the apex.
Lines of symmetry
If there is at least one line that divides a two-dimensional shape into two identical halves, the shape is said to have reflection symmetry, or line symmetry. The line that divides the shape into two matching halves is called the line of symmetry. This line can run in any direction. (A three-dimensional shape needs to be divided in half by a plane of symmetry rather than a line of symmetry, but this is not covered here.) Shapes can have more than one line of symmetry. In a regular polygon, the number of lines of symmetry is equal to the number of sides.
Look at the examples of shapes with reflection symmetry below. The dashed lines are the lines of symmetry. Notice that the triangle has three lines of symmetry, while the pentagon has five:
\(\) Check your understanding of lines of symmetry by completing one or both of the following activities:
Seven, as the number of lines of symmetry is equal to the number of sides.
A circle has infinitely many lines of symmetry, as any straight line passing through the centre is a line of symmetry.