Mathematics (9260) | Mensuration and calculation

Welcome to Mensuration and Calculation!

Hello everyone! In this chapter, we are diving into Mensuration. Don't let the big word scare you! Mensuration simply means the mathematics of measurement. We are going to learn how to calculate the size of things—specifically, the length, area, and volume of shapes and objects.

Why is this important? Whether you are calculating how much paint you need for a wall, how much concrete is required for a foundation, or the capacity of a water tank, these skills are essential in the real world. Think of this chapter as your mathematical toolkit for construction, design, and planning!

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Section 1: Measuring 2D Shapes – Perimeter and Area

A 2D shape is a flat shape (like a drawing on a piece of paper). We measure two things for 2D shapes: Perimeter and Area.

1.1 Perimeter (P): Walking the Edge

The Perimeter is the total distance around the outside of a shape. It's like walking along the boundary of a field.

• Calculation: Always add up the lengths of all the sides.
• Units: Length units (cm, m, km).

Example: If a rectangular garden is 5m long and 3m wide, the perimeter is 5 + 3 + 5 + 3 = 16m.

1.2 Area (A): Covering the Space

The Area is the amount of surface inside the boundary of the shape. This tells you how much carpet or paint you need.

• Units: Square units (cm², m², km²).

Area Formulas for Common Shapes:

1. Rectangle / Square

Formula: \(A = \text{length} \times \text{width}\)
(L x W)

2. Triangle

The area of a triangle is half the area of the rectangle it fits into. Remember, the height (\(h\)) must be perpendicular (at 90°) to the base (\(b\)).

Formula: \(A = \frac{1}{2} \times \text{base} \times \text{height}\)
(A = \(\frac{1}{2} b h\))

3. Parallelogram

A parallelogram is a 'slanted' rectangle. If you chop off the triangle on one end and move it to the other, it becomes a rectangle.

Formula: \(A = \text{base} \times \text{perpendicular height}\)
(A = \(b h\))

4. Trapezium (Trapezoid)

A trapezium has one pair of parallel sides (let’s call them \(a\) and \(b\)). We average the length of the parallel sides and multiply by the height.

Step-by-Step for Trapezium Area:
1. Add the parallel sides (\(a + b\)).
2. Divide that sum by 2 (this finds the average length).
3. Multiply by the height (\(h\)).

Formula: \(A = \frac{1}{2} (a + b) h\)
(A = \(\frac{a+b}{2} h\))

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Section 2: Circles – The Special Shape

Circles require special formulas because they don't have straight sides. We use the constant Pi (\(\pi\)) in all circle calculations. In GCSE exams, you usually use the \(\pi\) button on your calculator (approximately 3.14159...).

Prerequisite Check:
Radius (r): Distance from the centre to the edge.
Diameter (d): Distance across the circle through the centre. (\(d = 2r\))

2.1 Circumference (C) – Perimeter of a Circle

The Circumference is the perimeter (the length around the circle).

Formula (using Diameter): \(C = \pi d\)

Formula (using Radius): \(C = 2 \pi r\)

2.2 Area of a Circle (A)

The area is the space inside the circle.

Formula: \(A = \pi r^2\)

Memory Trick (Mnemonic):
Cherry Pie Delicious: \(C = \pi d\) (Circumference)
Apple Pies Are Too: \(A = \pi r^2\) (Area)

2.3 Sectors and Arcs

Sometimes, you only need to find the length or area of a part of a circle.
• A Sector is a slice (like a pizza slice).
• An Arc is the curved edge of that slice.

We calculate these by finding the fraction of the whole circle, which is determined by the angle \(\theta\).

1. Arc Length
We take the fraction of the Circumference:

Formula: \(\text{Arc Length} = \frac{\theta}{360} \times 2 \pi r\)

2. Sector Area
We take the fraction of the whole Area:

Formula: \(\text{Sector Area} = \frac{\theta}{360} \times \pi r^2\)

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Section 3: 3D Shapes – Calculating Volume

When we move to 3D shapes (like boxes or cans), we measure the space they fill. This is called Volume. Volume is measured in cube units (cm³, m³).

3.1 Prisms (Shapes with a Constant Cross-Section)

A Prism is any 3D shape that has the same cross-sectional shape throughout its length. Think of slicing a loaf of bread—each slice is identical.

The Universal Prism Volume Formula:

\(V = \text{Area of Cross-Section} \times \text{Length}\)

Specific Prism Formulas:

1. Cuboid (Rectangular Prism)

Cross-Section is a rectangle (Area = L x W).
Formula: \(V = l \times w \times h\)

2. Cylinder

Cross-Section is a circle (Area = \(\pi r^2\)).
Formula: \(V = \pi r^2 h\)

Encouragement: If you can find the area of the 2D base shape, finding the volume of the prism is usually just one extra step (multiplying by length or height)!

3.2 Pyramids and Cones (Shapes that Taper)

Pyramids (square base, triangular sides) and Cones (circular base) taper to a single point (an apex). Because of this, their volume is much smaller than the equivalent prism or cylinder. In fact, it's exactly one-third (1/3).

The Universal Tapering Volume Formula:

\(V = \frac{1}{3} \times \text{Base Area} \times \text{Perpendicular Height}\)

Specific Tapering Formulas:

1. Cone

Base Area is \(\pi r^2\).
Formula: \(V = \frac{1}{3} \pi r^2 h\)

2. Square/Rectangular Pyramid

Base Area is \(l \times w\).
Formula: \(V = \frac{1}{3} l w h\)

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Section 4: Surface Area (SA)

The Surface Area (SA) of a 3D object is the total area of all its outer faces combined. Think of it as the amount of wrapping paper needed to cover the object.

4.1 Surface Area of a Cuboid

A cuboid has 6 faces: a front and back, a top and bottom, and two sides. Since opposite faces are identical, you can calculate the area of three unique faces and double the result.

Step-by-Step Cuboid SA:
1. Calculate Area 1 (Front/Back): \(l \times h\)
2. Calculate Area 2 (Top/Bottom): \(l \times w\)
3. Calculate Area 3 (Side/Side): \(w \times h\)
4. Total SA = \(2(\text{Area 1}) + 2(\text{Area 2}) + 2(\text{Area 3})\)

4.2 Surface Area of a Cylinder

A cylinder is made up of two parts: the circular top and bottom, and the curved side.

If you unroll the curved side, it becomes a rectangle!

• The rectangle's width is the height (\(h\)) of the cylinder.
• The rectangle's length is the Circumference of the base (\(2 \pi r\)).

1. Area of Circular Ends: \(2 \times (\pi r^2)\)

2. Area of Curved Face: \(\text{Circumference} \times \text{Height} = 2 \pi r h\)

Total Cylinder SA Formula:

\(SA = 2 \pi r^2 + 2 \pi r h\)

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Section 5: Units and Practical Calculations

Mensuration often involves converting units, especially when dealing with area and volume, which can be tricky!

5.1 Converting Lengths, Areas, and Volumes

Remember the basic linear conversion: 1 metre (m) = 100 centimetres (cm).

1. Area Conversion (Square Units)

To convert square units, you must square the conversion factor (100).
\(1 \text{ m}^2 = 100 \times 100 \text{ cm}^2 = 10,000 \text{ cm}^2\)

If you are building a patio 2 m² in size, you need 20,000 cm² of slabs.

2. Volume Conversion (Cubic Units)

To convert cubic units, you must cube the conversion factor (100).
\(1 \text{ m}^3 = 100 \times 100 \times 100 \text{ cm}^3 = 1,000,000 \text{ cm}^3\)

5.2 Volume and Capacity (Litres)

Capacity often refers to how much liquid a container can hold, and it uses standard units like litres. You need to know the crucial conversion between volume (cm³) and capacity (litres):

1 Litre (\(L\)) = 1000 cubic centimetres (\(\text{cm}^3\))

Did you know? A carton of milk is usually 1 litre, meaning it holds 1000 cm³ of liquid.

Quick Tip for Litres:
To convert volume in cm³ to Litres, divide by 1000.
To convert volume in m³ to Litres, multiply by 1000 (since 1 m³ = 1000 L).

You have covered all the essential measurement techniques needed for your GCSE! Keep practicing those formulas, and you'll master this chapter in no time!