Welcome to Mensuration: Area and Perimeter
Hi everyone! Mensuration is all about measuring the size of things in the real world. In this chapter, we focus on two core measurements for 2D shapes: Area and Perimeter.
Think of it this way:
- Perimeter: If you were fencing a field, the perimeter is the total length of the fence you need. It’s the measure of the boundary.
- Area: If you were covering that field with grass seed, the area is the amount of surface inside the boundary.
Mastering these calculations is essential for your IGCSE exams and for practical applications like building, DIY, or even calculating the amount of paint you need for a wall! Let's break it down.
Section 1: Units and Basic Concepts
Understanding the Difference
The biggest mistake students make is mixing up perimeter and area, especially with the units!
Key Definitions and Units (C6.1)
- Perimeter: The total distance around the outside of a shape.
Units: Measured in standard length units (mm, cm, m, km). - Area: The amount of surface covered by the shape.
Units: Measured in squared units (mm², cm², m², km²).
Quick Review: Units (C6.1)
Always check that all lengths are in the same units before you start calculating! For example, you cannot calculate the area of a rectangle using length in metres (m) and width in centimetres (cm).
Example: To convert \(1\,m^2\) to \(cm^2\):
Since \(1\,m = 100\,cm\), then \(1\,m^2 = 100\,cm \times 100\,cm = 10,000\,cm^2\). Be careful with squared units!
Key Takeaway for Section 1
Perimeter is length (single units); Area is surface (squared units).
Section 2: Area and Perimeter of Standard Shapes (C6.2)
For perimeter, you simply add up the lengths of all the exterior sides. The challenge usually lies in remembering the correct formula for the area.
Note: You need to memorise the formulas for the area of the trapezium, parallelogram, and rectangle. The formula for the area of a triangle (\(A = \frac{1}{2}bh\)) is provided in the formula list.
1. Rectangle and Square
If the rectangle has length \(L\) and width \(W\):
- Perimeter (P): \(P = 2L + 2W\)
- Area (A): \(A = L \times W\)
2. Triangle
The area calculation for a triangle is crucial. You must use the perpendicular height.
Important! The height (\(h\)) must be measured at a right angle (90°) to the base (\(b\)).
- Perimeter (P): \(P = \text{Sum of all three sides}\)
- Area (A): \(A = \frac{1}{2} \times \text{base} \times \text{height} \quad \left(A = \frac{1}{2}bh\right)\)
Did you know? The formula works even if the height line falls outside the triangle (for obtuse triangles).
3. Parallelogram
A parallelogram is a 'tilted' rectangle. If you cut off the triangle from one side and move it to the other, it becomes a rectangle!
Therefore, its area formula is very similar to a rectangle, but again, you must use the perpendicular height.
- Perimeter (P): \(P = 2 \times (\text{side A} + \text{side B})\)
- Area (A): \(A = \text{base} \times \text{perpendicular height} \quad \left(A = bh\right)\)
4. Trapezium (Trapezoid)
A trapezium is a quadrilateral with exactly one pair of parallel sides (let's call them \(a\) and \(b\)).
The area formula averages the two parallel sides and multiplies by the perpendicular height (\(h\)).
- Perimeter (P): \(P = \text{Sum of all four sides}\)
- Area (A): \(A = \frac{1}{2} \times (\text{sum of parallel sides}) \times \text{height} \quad \left(A = \frac{1}{2}(a+b)h\right)\)
Memory Aid for Trapezium: "Averaging the parallel sides gives you the middle length, then multiply by height."
Key Takeaway for Section 2
Always identify the perpendicular height for parallelograms and triangles. Remember the trapezium formula!
Section 3: Calculations Involving Circles (C6.3)
Calculations involving circles always use the number Pi (\(\pi\)). You should use the \(\pi\) button on your calculator for accuracy, or the value 3.142 if specified.
Key Circle Vocabulary
- Radius (\(r\)): Distance from the centre to the edge.
- Diameter (\(d\)): Distance across the circle through the centre (\(d = 2r\)).
- Circumference (\(C\)): The perimeter (distance around) the circle.
The following formulas for circles are given in your exam formula list:
1. Circumference (Perimeter)
The distance around the circle.
\(C = \pi d\)
OR
\(C = 2\pi r\)
2. Area of a Circle
The space enclosed by the circle.
\(A = \pi r^2\)
Common Mistake Alert!
When calculating the area, check if you are given the radius (\(r\)) or the diameter (\(d\)). If given the diameter, remember to halve it first before using \(A = \pi r^2\).
Step-by-Step Example: Find the area of a circle with a diameter of 10 cm.
- Find the radius: \(r = 10 \div 2 = 5\,cm\)
- Apply the area formula: \(A = \pi (5)^2\)
- \(A = 25\pi \approx 78.5\,cm^2\) (to 3 s.f.)
Key Takeaway for Section 3
Circumference uses radius/diameter once (\(2\pi r\)), Area uses radius squared (\(\pi r^2\)). Always check for \(r\) vs. \(d\).
Section 4: Arcs and Sectors (C6.3)
Arcs and sectors are simply portions of a full circle. To find their measurements, we calculate what fraction of the full circle they represent using the angle at the centre (\(\theta\)).
The Key Idea: The Fraction
Since a full circle is \(360^\circ\), the fraction of the circle you are working with is always: \(\frac{\theta}{360}\)
1. Arc Length
The Arc Length (\(L\)) is the length of the curved edge of the sector (a part of the circumference).
Formula (given in exam):
\(L = \frac{\theta}{360} \times 2\pi r\)
Analogy: You are finding the total distance around the edge (\(2\pi r\)), but only for the slice defined by the angle \(\theta\).
2. Sector Area
The Sector Area (\(A\)) is the area of the 'pizza slice' (a part of the circle's total area).
Formula (given in exam):
\(A = \frac{\theta}{360} \times \pi r^2\)
Example: Calculate the area of a sector with radius 8 cm and a central angle of \(45^\circ\).
\(A = \frac{45}{360} \times \pi (8)^2\)
\(A = \frac{1}{8} \times 64\pi\)
\(A = 8\pi \approx 25.1\,cm^2\)
3. Perimeter of a Sector
This is often asked in problems involving arcs and sectors, but is NOT a formula you are given!
The perimeter of a sector includes the curved arc length AND the two straight radii that connect the arc to the centre.
Perimeter of Sector \(P = \text{Arc Length} + r + r\)
\(P = \left(\frac{\theta}{360} \times 2\pi r\right) + 2r\)
Key Takeaway for Section 4
Arc length and sector area are found by multiplying the fraction \(\frac{\theta}{360}\) by the corresponding whole circle formula. Don't forget the radii when finding the perimeter of a sector!
Section 5: Compound Shapes and Parts of Shapes (C6.5)
A compound shape is simply a shape made up of two or more standard geometrical shapes (like a rectangle and a semicircle stuck together).
1. Calculating Compound Area
The strategy here is to decompose the complex shape into simpler parts (Sectors, Triangles, Rectangles, etc.).
Step-by-Step Method:
- Divide: Break the complex shape into standard, non-overlapping shapes.
- Calculate Parts: Find the area of each individual standard shape using the correct formulas.
- Combine: Add (or sometimes subtract) the areas together to get the total area.
Example: A shape is made of a rectangle (4m by 6m) and a triangle (base 6m, height 3m) attached to the 6m side.
- Area of Rectangle: \(4 \times 6 = 24\,m^2\)
- Area of Triangle: \(\frac{1}{2} \times 6 \times 3 = 9\,m^2\)
- Total Area: \(24 + 9 = 33\,m^2\)
2. Calculating Compound Perimeter
This is where students must be most careful!
Critical Rule: The perimeter is only the boundary of the figure. Do not include any internal lines used to divide the shape.
Example: Using the shape above (rectangle 4x6, triangle base 6), the 6m side that joins the two shapes is an internal line. It is NOT part of the perimeter.
Perimeter Method:
- Trace the Boundary: Imagine drawing the shape without lifting your pen.
- Identify External Lengths: List every length that is on the outside edge. This might involve calculating arc lengths or diagonal lengths (using Pythagoras' theorem).
- Sum: Add only these external lengths.
Common Mistake to Avoid!
If you have a composite shape involving a semicircle, remember that the curved edge is an arc length (\(\frac{1}{2} \times 2\pi r\)), but the straight diameter (the joining line) is not part of the final perimeter.
Key Takeaway for Section 5
Area = Add (or subtract) the individual shapes. Perimeter = Only measure the outside edges!