Geometry and Trigonometry (M4)

In geometry, angles are usually measured in degrees, where a rotation of represents a complete turn.

If a complete turn is split into several angles, those angles must add up to

A straight line splits a complete turn in half. Therefore, the angles on one side of a straight line add up to

You can use these rules to find missing angles around a point, or on a straight line.

It is important to know the sum of the interior angles of a triangle and a quadrilateral.

• Any shape with three sides is a triangle. There are also three interior angles between these sides. These angles always add up to .

• Any shape with four sides is a quadrilateral. There are also four interior angles between these sides. These angles always add up to .

It is useful to know that the sum of the interior angles of any simple polygon is equal to the number of sides in the shape minus 2, multiplied by :

For example, for the angles in the triangle:

For the angles in the kite (a type of quadrilateral):

When two angles are on the same side of a straight line, like and below, they add up to Therefore, when two lines cross at a point, you can easily use one of the angles at this point to find all the other angles.

Furthermore, if a line crosses two parallel lines, the angles on both parallel lines are equal.

By following these principles:

• because they are on parallel lines
• because they are on parallel lines

Hence, and are equal.

Furthermore,

By a similar reasoning, and are equal.

If any of these eight angles is known, we can easily calculate any of the others.

To find the area of a rectangle (or a square), multiply its base by its height:

To find the area of a triangle, multiply its base by its height, and halve the answer. This is because the triangle fills exactly half of a rectangle with the same base and height:

The circumference of a circle is the length of its perimeter. To calculate the circumference and area of a circle, you need to use the value of (the Greek letter pi).

Your calculator has a button for this value.

The radius of a circle is the distance from its centre to anywhere on its perimeter.

The diameter of a circle is the straight-line distance passing through its centre, connecting two opposite points on the circumference. It is therefore twice the radius.

You can use the formulas:

• Circumference: or
• Area:

In this particular circle:

(Results are rounded to 3 significant figures.)

For some shapes, such as rectangles, triangles and circles, you can use a formula to find the area.

Some shapes have a more complex structure, but you might find a way to split them into several simple shapes whose area you can easily calculate.

For example, the area of the front wall of this barn can be considered a large rectangle, plus the triangular roof, minus the rectangular door.

The surface area of a 3D shape is the total area of all its exterior surfaces.

The surface area of some shapes, such as cuboids (and cubes), consists of several flat surfaces. For shapes like these, you can simply find these areas and add them together.

For example, consider a cuboid with dimensions and

The cuboid has:

• A front and back (yellow), whose areas are each
• Two side (red), whose areas are each
• A top and bottom (blue), whose areas are each

Add these together:

The surface of a cylinder consists of two circles (at the ends) and a curved surface.

If you flattened the curved surface, it would be a rectangle where:

• Length is the length of the cylinder.
• Height is the circumference of the circles at the ends.

For this cylinder:

It is useful to note that for a cylinder of length, and radius,

In mathematics, a prism is defined as any 3D shape formed from a 2D shape that is given depth.

You need to calculate the volumes of the following prisms:

• cuboids (and cubes), where the 2D shape is a rectangle (or a square), and
• cylinders, where the 2D shape is a circle.

For any prism, the volume is calculated as the area of the 2D shape, multiplied by the depth.

You need to know the formulas for the surface area and volume of a sphere of radius, :

For this sphere:

In any triangle, the sum of angles is always . This means that if you know any two of the angles, you can easily calculate the other angle.

Here, the sum of the known angles is .

The sum of all angles must be .

, so the missing angle is .

In a right-angled triangle, the right angle is always the largest angle. The side opposite the right angle is called the hypotenuse, which is guaranteed to be the longest side.

In a right-angled triangle, the lengths of the sides always satisfy Pythagoras’ Theorem:

You may see this written as:

Where:

• represents the hypotenuse, and
• and are the other sides.

For this triangle:

This illustrates Pythagoras’ Theorem for this triangle.

All triangles consist of three sides and three angles. If you know any three of these six measurements, you can typically determine the length of the remaining sides and the measure of the remaining angles.

It is important to know the following facts and techniques to determine the lengths and angles of a triangle:

• Angles in a triangle add up to
• Pythagoras’ Theorem:

• Basic trigonometry: sin, cos, and tan
• Inverse trigonometry: and

• Sine rule:
• Cosine rule:

Suppose you have a triangle and you resize it to make a bigger or smaller triangle. These triangles have the same angles. You can describe these triangles as similar triangles.

It is important to note that the ratio between any two side lengths is the same in both triangles.

In the larger triangle, all lengths are double. Therefore, both triangles have the same shape, and their angles are the same. The ratio between any two sides, therefore, is also the same.

For example, consider the ratio between the shortest and longest sides.

• Small triangle:

• Large triangle:

The trigonometric ratios of sin, cos, and tan are also ratios between side lengths. Therefore, they depend on the angle, but not on the size of the triangle.

In any right-angled triangle, you can use the trigonometric ratios of sin, cos, and tan to find missing side lengths and missing angles.

It is important to know the following definitions, because they are the basic formulas of trigonometry:

Some students use the phrase to help them remember these definitions, where, for example, “SOH” is the first letter of each word in the first formula.

You can use trigonometry to find missing angles. Typically, you’ll end up with an equation like:

To solve these equations, you need to use the inverse trigonometric functions, which are the opposites of the functions sin, cos, and tan. In this case:

The table below presents the three basic trigonometric functions and their inverses, along with illustrative examples in parentheses.

It is important to familiarise yourself with your calculator’s functions to avoid wasting time during the exam.

The sine rule is a remarkable mathematical fact that allows you to find unknown angles and lengths in triangles that do not have a right angle.

The sine rule says that for any triangle, the ratio:

is the same for each of its three angles.

The sine rule is often written as:

Where is the side opposite angle and is the side opposite angle

When given two side lengths and an angle not included between those sides of an unknown triangle, there might be two possible triangles that can be drawn from the incomplete description.

This is the ‘problem with angle–side–side’.

For example, suppose we know angle and lengths and as shown below.

There are two triangles matching this description. There is no way to know which one is correct!

In the blue triangle:

In the green triangle:

• When applying the sine rule to find a missing angle, this arises because takes the same value for two angles in the range

• When applying the cosine rule to find a missing length, this might arise when this rule gives a quadratic equation with two solutions.

The cosine rule is an extension to Pythagoras’ Theorem that works for all triangles, even if they do not have a right angle.

In a right-angled triangle, the hypotenuse, depends on the lengths and according to Pythagoras’ Theorem:

In general, the unknown length, varies depending on the opposite angle, according to the cosine rule:

It is useful to note that in a right-angled triangle, In this case, the cosine rule matches Pythagoras’ Theorem.

Sometimes, you may have information about a triangle that cannot exist, because the lengths and angles are impossible.

One case of this is if the longest side is longer than the sum of the shortest sides. This is always impossible.

Suppose you calculate the largest angle in a triangle with lengths 30, 40 and 85.

You must use the cosine rule:

There are no real solutions to this, as cosine has a minimum value of –1. Therefore, the triangle is impossible. Indeed, so the longest side is too long to fit.

Sometimes, you may have information about a triangle that cannot exist, because the lengths and angles are impossible.

If you know the lengths of two sides of a triangle and one of the angles that is not between those sides, it is sometimes impossible for a triangle to match this description.

Suppose a triangle contains an angle of opposite a side of length 30. One of the other sides has a length of 50. What could be the angle opposite this side?

In fact, the side of length 30 is not long enough to reach the other side!

Using the sine rule:

This is impossible, because the maximum value of is 1.

If using the cosine rule to find one of the unknown angles, you will end up with a quadratic equation with no real roots, also indicating that the triangle is impossible.

You should be aware of the following useful small angle approximations, which are only valid in radians (not degrees), and only when the angle is close to zero.

For example, you can use the approximations when the angle is 0.2 radians.

In certain circumstances, you may be asked to recall and use these approximations. Exam questions will indicate if an angle is “small” enough.

For example, when deriving formulas to describe Young’s double-slit experiment in Physics, you may need the approximation

You should be aware of the following useful small angle approximation, which is only valid when the angle is close to zero.

For example, you can use this approximation when the angle is 0.2 radians.

It is useful to know that a more precise approximation exists (although only in radians), but you won’t need to remember this for your exam:

In certain circumstances, you may be asked to recall and use the approximation that Exam questions will indicate if an angle is “small” enough.

A full turn can be divided into 360 degrees, written Degrees are used to measure angles and rotation.

A full turn can instead be divided into other quantities. One such quantity is the radian, which is defined as the angle in a circle subtended by an arc of the same length as the circle’s radius, as illustrated below.

The circumference of a circle is , so radians make a full turn:

It is important to note that radians are a key concept in mathematics and science. A solid understanding of them is essential for your exams and further learning.

A full turn can be divided into or radians. Therefore:

Consequently, the conversion between radians and degrees is:

You need to remember this conversion for your exams.

For example, to describe an angle of 2.3 radians in degrees:

For example, to describe an angle of in radians:

Angles in radians are often written as a fraction of It is useful to note some common angles in radians.

However, you can always calculate these (and any other angle) easily using: