Mensuration and trigonometry Revision Notes

What is Scale?

Scale refers to the relationship between the distance on a map or drawing and the corresponding actual distance on the ground or in real life. It allows large areas or objects to be represented compactly. Understanding scale is fundamental for interpreting any map or technical drawing. Without scale, a drawing is just an illustration, not a precise representation. Always look for the scale information provided.

Types of Scale Representation

Scales are typically expressed in three main ways:

1. Ratio Scale: Expressed as a ratio, e.g., 1:1000. This means 1 unit on the map represents 1000 of the same units in reality. For example, 1 cm on the map is 1000 cm (or 10 m) in reality.
2. Statement Scale: Expressed as a verbal statement, e.g., '1 cm represents 5 km'. This directly tells you the equivalence between map and real-world distances, often using different units for clarity.
3. Graphic Scale Bar: A visual bar on the map marked with real-world distances (e.g., 0-100 m). You can use a ruler to measure the bar and then apply that measurement to other features on the map. This type of scale remains accurate even if the map is resized.

Calculating Actual Distance from Map Distance

To find the actual distance using a map:

1. Measure the distance on the map using a ruler.
2. Identify the scale. If it's a ratio (e.g., 1:5000), multiply your measured map distance by the second number in the ratio.
3. If the scale is a statement (e.g., '1 cm represents 2 km'), multiply your measured map distance by the real-world distance indicated (e.g., 2 km).
4. Ensure all units are consistent or convert them to appropriate practical units (e.g., cm to m, or km). For example, if a map scale is 1:2000 and you measure 3 cm, the actual distance is 3 cm * 2000 = 6000 cm = 60 m.

Calculating Map Distance from Actual Distance

To represent a real-world distance on a map:

1. Know the actual distance you want to represent.
2. Identify the desired map scale (e.g., 1:500).
3. Convert the actual distance to the same units as the map measurement (e.g., convert 10 metres to 1000 cm).
4. Divide the actual distance by the scale factor. For example, to represent 10 m (1000 cm) on a map with a 1:500 scale, the map distance would be 1000 cm / 500 = 2 cm. Always pay close attention to unit conversions before calculation.

What is Perimeter?

The perimeter is the total distance around the outside of a shape. Think of it as outlining the shape with a string and then measuring the length of that string. For any closed two-dimensional figure, the perimeter is found by summing the lengths of all its boundary edges. Units for perimeter are always units of length, e.g., mm, cm, m, km.

Recognising Rectilinear Shapes

Rectilinear shapes are polygons where all sides meet at right angles (90 degrees). Common examples include squares, rectangles, and L-shaped figures. These shapes are essentially made up of interconnected straight lines. Because all angles are right angles, you can easily determine missing side lengths by adding or subtracting known lengths along parallel lines.

Calculating Perimeter of Rectilinear Shapes

To find the perimeter, you must add the lengths of ALL the outer sides of the shape. If some side lengths are not given, particularly in complex rectilinear shapes (like an L-shape), you can deduce them. For instance, in an L-shape, if you have a horizontal 'total' length, and a shorter horizontal segment is given, the remaining horizontal segment can be found by subtracting. The same applies to vertical segments. Always double-check that you have included every external side in your sum.

Example: L-Shaped Perimeter

Consider an L-shaped room. You need to fence it. The room has six sides. If the outer dimensions are 8m by 6m, and an internal cut-out is 3m by 2m, you would have sides of 8m, 6m, 5m (8-3), 2m, 3m, 4m (6-2). Adding these: 8 + 6 + 5 + 2 + 3 + 4 = 28m. The sum of the horizontal segments must equal the overall horizontal length, and similarly for vertical segments. Be careful not to include any internal lines in your calculation.

What is Circumference?

The circumference is the total distance around the outside edge of a circle. Think of it as the perimeter of a circle. It's a fundamental property of circles and is always a certain proportion of its size. Understanding this concept is key to solving problems involving circular shapes in geometry and real-world applications. It's measured in units of length, such as centimetres (cm) or metres (m).

Key Terms: Diameter and Radius

There are two essential measurements for a circle: the diameter and the radius. The diameter (d) is a straight line segment that passes through the centre of the circle and connects two points on its edge. The radius (r) is a straight line segment from the centre of the circle to any point on its edge. Critically, the diameter is always exactly twice the length of the radius (d = 2r). This relationship is central to using the correct formula.

The Constant Pi (π)

Pi (π) is a special mathematical constant, approximately 3.14159. It represents the ratio of a circle's circumference to its diameter. No matter how big or small a circle is, if you divide its circumference by its diameter, you will always get π. For most calculations, you can use the π button on your calculator for greater accuracy. If you need to give an approximate answer, you might be told to use π ≈ 3.14 or π ≈ 22/7.

Circumference Formulas

There are two main formulas for calculating the circumference (C) of a circle, depending on whether you are given the diameter or the radius:

1. If you know the diameter (d): C = πd
2. If you know the radius (r): C = 2πr

Both formulas are equivalent because the diameter is twice the radius (d = 2r). For example, if a circle has a radius of 5 cm, its diameter is 10 cm. Using C = πd, C = π * 10 = 10π cm. Using C = 2πr, C = 2 * π * 5 = 10π cm. Always remember to include the correct units in your final answer.

What is Circumference?

The circumference of a circle is the distance around its outer edge. It's like the perimeter but for a circular shape. Imagine cutting a string exactly the length of the circle's edge and then straightening it out; that length is the circumference. It is measured in units of length, such as centimetres (cm), metres (m), or kilometres (km).

Key Components of a Circle

To calculate circumference, you need to know about the diameter and radius. The 'diameter' (d) is a straight line that goes from one side of the circle to the other, passing directly through the centre. The 'radius' (r) is a straight line from the centre of the circle to any point on its circumference. The diameter is always exactly twice the length of the radius (d = 2r), meaning the radius is half the diameter (r = d/2).

The Role of Pi (π)

Pi (π) is a special mathematical constant, approximately equal to 3.14159. It represents the ratio of a circle's circumference to its diameter. No matter how big or small a circle is, if you divide its circumference by its diameter, you'll always get π. In calculations, you can use π as a symbol for exact answers, or approximate values like 3.14 or 22/7 if specified or for practical calculations.

Circumference Formulas

There are two main formulas for calculating the circumference (C) of a circle:

1. If you know the diameter (d): C = πd
2. If you know the radius (r): C = 2πr

Both formulas achieve the same result because 'd' is equal to '2r'. Choose the formula that uses the information you are given in the problem. Always remember to use the correct units in your final answer.