Algebra Revision Notes

Understanding the Goal

Changing the subject of a formula means to rearrange an equation so that a different specified variable is isolated on one side of the equals sign. This new form of the equation allows you to calculate the value of the new subject directly if you know the values of all other variables. The process is similar to solving equations, but instead of finding a numerical value, you are algebraically manipulating the expression.

Inverse Operations and Order

To isolate the desired variable, you must apply inverse operations. Think of operations applied to the subject in reverse order of BIDMAS/PEMDAS. For example, if a variable is squared, then multiplied by a constant, then has another term added, you would first subtract the added term, then divide by the constant, and finally take the square root. Always perform the inverse operation to 'undo' the current operation on the subject.

Dealing with Fractions and Roots

When the subject is part of a fraction, multiply both sides by the denominator first to eliminate the fraction. If the subject is in the denominator, you'll need to multiply by it to bring it to the numerator, then divide by the other terms to isolate it. For terms involving square roots (or other roots), square (or raise to the appropriate power) both sides of the equation. If the subject is squared, take the square root of both sides. Remember to consider both positive and negative roots where appropriate, though often in formulas, physical constraints limit to the positive root.

Subject Appearing Multiple Times

If the variable you want to make the subject appears in more than one term in the equation, the strategy changes slightly. First, collect all terms containing the target variable on one side of the equation and all other terms on the opposite side. Then, factorise out the common target variable. Once factorised, you will have the variable multiplied by an expression in brackets, which you can then divide by to isolate the variable. This technique is often encountered in higher-level problems.

What is Circumference?

The circumference of a circle is the total distance around its curved edge. Think of it as the 'perimeter' of a circle. It can be calculated using one of two interchangeable formulas, depending on whether the radius or diameter is known. The constant π (pi) is essential for these calculations, representing the ratio of a circle's circumference to its diameter, approximately 3.14159. For accuracy, it's often best to use the π button on your calculator or leave answers in terms of π if specified.

Circumference Formulas: C = 2πr or C = πd

There are two main formulas for the circumference of a circle: C = 2πr, where 'r' is the radius (distance from the centre to the edge). C = πd, where 'd' is the diameter (distance across the circle through the centre). It's important to note that the diameter is always twice the radius (d = 2r). Therefore, 2πr is exactly the same as π(2r) which is πd. You can use whichever formula is most convenient based on the information given in a problem.

What is the Area of a Circle?

The area of a circle refers to the amount of flat space enclosed within its circumference. Unlike circumference, which is a linear measurement, area is a two-dimensional measurement and is expressed in square units (e.g., cm², m²). The area calculation also relies on the radius of the circle and the constant π. Understanding this formula is crucial for solving problems involving the space a circle occupies.

Area Formula: A = πr²

The formula for the area of a circle is A = πr², where 'r' is the radius of the circle. This formula highlights that the area depends on the square of the radius, meaning that small changes in the radius can lead to significant changes in the area. Always ensure you are using the radius in this formula. If you are given the diameter, you must first halve it to find the radius before applying the area formula.

Circumference Formulae

The circumference (C) is the distance around the outside of a circle. There are two formulae you can use, depending on whether you are given the radius or the diameter:

1. C = 2πr (where 'r' is the radius)
2. C = πd (where 'd' is the diameter)

Remember that the diameter is always twice the radius (d = 2r), so these two formulae are equivalent. For example, if a circle has a radius of 5 cm, its circumference is C = 2 * π * 5 = 10π cm. If its diameter is 10 cm, its circumference is C = π * 10 = 10π cm.

Area Formula

The area (A) is the amount of space inside a circle. The formula for the area of a circle uses the radius:

1. A = πr² (where 'r' is the radius)

It is crucial to remember that the area formula specifically uses the radius, squared. If you are given the diameter, you must first calculate the radius by halving the diameter before you can find the area. For example, if a circle has a radius of 5 cm, its area is A = π * 5² = 25π cm². If its diameter is 10 cm, you first find the radius (10 / 2 = 5 cm), then calculate the area.

Understanding π (Pi)

Pi (π) is a mathematical constant approximately equal to 3.14159265... It represents the ratio of a circle's circumference to its diameter. In calculations, you may be asked to:

• Leave your answer 'in terms of π'. This means your answer will still contain the π symbol, e.g., 10π cm.
• Use the π button on your calculator for a more accurate answer, then round to a specified number of decimal places or significant figures.
• Use an approximation like 3.14 or 22/7, though this is less common at GCSE level unless specified. Always follow the instructions in the question.

Key Definitions: Radius and Diameter

The radius (r) of a circle is the distance from the centre of the circle to any point on its circumference. It's a line segment. The diameter (d) of a circle is the distance across the circle passing through its centre. It's also a line segment. The diameter is always exactly twice the length of the radius (d = 2r). Therefore, if you know the diameter, you can find the radius by dividing by 2 (r = d/2).

Circumference of a Circle: C = 2πr or C = πd

The circumference is the total distance around the edge of a circle. You have two primary formulae to calculate it. The first, C = 2πr, uses the radius (r), which is the distance from the centre to any point on the edge. The second, C = πd, uses the diameter (d), which is the distance across the circle passing through its centre. Remember that the diameter is always double the radius (d = 2r). Both formulae yield the same result. For example, if a circle has a radius of 5 cm, its circumference is C = 2 × π × 5 = 10π cm. If the diameter is 10 cm, then C = π × 10 = 10π cm. Pay attention to the units; circumference is a linear measurement.

Area of a Circle: A = πr²

The area of a circle is the amount of surface enclosed within its boundary. The formula for the area is A = πr², where 'r' is the radius of the circle. It is crucial to square the radius first, then multiply by π. Do not multiply the radius by 2. For instance, if the radius of a circle is 7 cm, its area is A = π × (7 cm)² = π × 49 cm² = 49π cm². If you are given the diameter instead of the radius, you must first halve the diameter to find the radius before applying the formula. For example, if the diameter is 12 cm, the radius is 6 cm, and the area is A = π × (6 cm)² = 36π cm². Always include appropriate square units for area, e.g., cm² or m².

Using Pi (π) in Calculations

In most exam questions, you can use the π button on your calculator for the highest level of accuracy. If a question asks for an answer 'in terms of π', you should leave π in your final answer (e.g., 10π cm). If a numerical answer is required, you will typically be asked to round to a specific number of decimal places or significant figures. If no rounding instruction is given, it is good practice to use the π button and then round to three significant figures unless otherwise indicated. Sometimes, a question might specify an approximation for π, such as 3.14 or 22/7; always follow these instructions precisely.