Mensuration and trigonometry Revision Notes

What is a Trapezium?

A trapezium is a 2D four-sided shape (quadrilateral) that has at least one pair of parallel sides. These parallel sides can be different lengths. The other two non-parallel sides can be of any length and can be slanted. Understanding that there must be at least one pair of parallel sides is key to identifying a trapezium correctly.

The Formula for Area

The area of a trapezium is calculated using the formula: Area = ½ × (a + b) × h. Here, 'a' and 'b' represent the lengths of the two parallel sides. It doesn't matter which parallel side you call 'a' and which you call 'b'. The 'h' represents the perpendicular height, which is the shortest distance between the two parallel sides. It's essential that 'h' is the perpendicular height, not the length of a non-parallel slanted side.

Applying the Formula Step-by-Step

To find the area:

1. Identify the two parallel sides and label their lengths as 'a' and 'b'.
2. Identify the perpendicular height between these parallel sides and label it 'h'.
3. Add the lengths of the parallel sides together: (a + b).
4. Multiply this sum by the perpendicular height: (a + b) × h.
5. Finally, divide the result by 2 (or multiply by ½): ½ × (a + b) × h.

Always include appropriate square units (e.g., cm², m²) in your final answer.

Common Errors to Avoid

A common mistake is using a non-perpendicular side length instead of the actual perpendicular height. Always look for the line that forms a 90-degree angle with both parallel sides. Another error is forgetting to divide by 2 at the end of the calculation. Remember the formula is for half of a parallelogram, so division by two is crucial. Ensure all measurements are in the same units before calculations.

Formula for Area of a Circle

The area of a circle (A) is calculated using the formula A = πr², where 'π' (pi) is a mathematical constant (approximately 3.14159...) and 'r' is the radius of the circle. The radius is the distance from the centre of the circle to its edge. You will often be asked to leave answers in terms of π or round to a certain number of decimal places or significant figures. Familiarise yourself with both approaches.

Radius vs. Diameter

It's crucial to distinguish between the radius and the diameter. The diameter (d) is the distance across the circle passing through its centre. The radius (r) is half of the diameter, meaning r = d/2. If a problem provides the diameter, your first step must be to calculate the radius before applying the area formula. For example, if the diameter is 10 cm, the radius is 10/2 = 5 cm.

Calculations and Units

When calculating the area, square the radius first, then multiply by π. Always remember to include the correct units for area, which are square units (e.g., cm², m², mm²). For example, if the radius is in metres, the area will be in metres squared (m²). Pay close attention to instructions regarding rounding; if no specific rounding is mentioned, leaving the answer in terms of π is often preferred for exactness.

The Formula for Area of a Circle

The area (A) of a circle is calculated using the formula A = πr², where 'r' is the radius of the circle. The constant π (pi) is an irrational number, approximately 3.14159. In calculations, you might be asked to leave your answer in terms of π or use 3.14, 3.142, or the π button on your calculator. Always check the required level of precision.

Radius vs. Diameter

It's crucial to distinguish between the radius and the diameter. The radius (r) is the distance from the centre of the circle to its edge. The diameter (d) is the distance across the circle passing through its centre, meaning d = 2r. If you are given the diameter, you must first divide it by 2 to find the radius before using the area formula. For example, if diameter = 14 cm, then radius = 14/2 = 7 cm.

Calculating Area with Given Radius

If you are given the radius directly, substitute its value into the formula A = πr². For instance, if r = 6 cm, then A = π × (6)² = 36π cm². If you need a numerical answer, use π ≈ 3.14: A ≈ 36 × 3.14 = 113.04 cm². Remember to include appropriate units, which will be square units (e.g., cm², m², mm²).

Calculating Area with Given Diameter

When given the diameter, the first step is to calculate the radius by dividing the diameter by 2. Then, proceed as if you were given the radius. For example, if a circle has a diameter of 20 mm, the radius is 20/2 = 10 mm. The area would then be A = π × (10)² = 100π mm². Using a calculator, this would be approximately 314.16 mm² (to two decimal places).

What are Composite Shapes?

Composite shapes are figures that can be divided into simpler, standard geometric shapes like rectangles, squares, triangles, or circles. Think of an 'L' shape: it's not a basic rectangle itself, but you can see it as two rectangles put together, or a larger rectangle with a smaller one cut out. Learning to break these down is the first step to finding their area.

Two Main Methods for Composite Area

There are two primary strategies to calculate the area of composite shapes. Method 1: Divide the composite shape into several simpler, non-overlapping shapes (e.g., two rectangles). Calculate the area of each smaller shape and then add these individual areas together to get the total area. Method 2: Envision the composite shape as a larger, simpler shape from which a smaller shape has been removed. Calculate the area of the larger shape and subtract the area of the 'removed' part to find the total area of the composite figure. Both methods should yield the same result if applied correctly.

Essential Area Formulas

To work with composite shapes, you need to remember the basic area formulas:

• Rectangle/Square: Area = length × width (or side × side for a square).
• Triangle: Area = 0.5 × base × height.
• Circle: Area = π × radius².

Make sure you can correctly identify the base, height, length, width, and radius for each component shape. For triangles, the height must be perpendicular to the base.

Step-by-Step Calculation Process

1. Analyze the shape: Look at the composite shape and decide how best to split it into simpler shapes or how to view it as a larger shape minus a smaller one.
2. Identify dimensions: Determine all necessary side lengths, bases, heights, or radii for the component shapes. You might need to infer some dimensions from the given ones.
3. Calculate individual areas: Apply the correct area formula to each basic shape.
4. Combine areas: Add the areas of the component shapes (if dividing) or subtract the 'removed' area from the total (if subtracting).
5. State units: Always include the correct square units (e.g., cm², m²) in your final answer.