Section B — Chapters 23–39
Applied Measure II
Junior Cycle — 1st Year
- ✓By the end of this lesson students will be able to calculate the perimeter and area of triangles.
- ✓By the end of this lesson students will be able to calculate the perimeter (circumference) and area of discs (circles).
- ✓By the end of this lesson students will be able to calculate the perimeter and area of composite 2D shapes.
- ✓By the end of this lesson students will be able to calculate the volume of rectangular solids (cuboids) and cylinders.
- ✓By the end of this lesson students will be able to interpret and use scaled diagrams to find actual measurements.
Key concepts
The perimeter of a 2D shape is the total distance around its outer boundary. It is measured in units of length (e.g., cm, m).
The area of a 2D shape is the amount of surface it covers. It is measured in square units (e.g., cm², m²).
A triangle is a polygon with three sides. To find its perimeter, add the lengths of all three sides. To find its area, multiply half of its base by its perpendicular height.
A disc (or circle) is a 2D shape where all points on its boundary are equidistant from a central point. The radius (r) is the distance from the centre to the boundary. The diameter (d) is the distance across the disc through the centre (d = 2r). The perimeter of a disc is called its circumference.
Composite shapes are made up of two or more basic geometric shapes (e.g., rectangles, triangles, discs). To find their perimeter, add the lengths of all the outer edges. To find their area, break the shape down into simpler shapes, calculate the area of each, and then add or subtract them as appropriate.
The volume of a 3D object is the amount of space it occupies. It is measured in cubic units (e.g., cm³, m³).
A rectangular solid (or cuboid) is a 3D shape with six rectangular faces. Its volume is found by multiplying its length, width, and height.
A cylinder is a 3D shape with two parallel circular bases and a curved surface. Its volume is found by multiplying the area of its circular base by its height.
A scaled diagram is a drawing where all measurements are proportionally reduced or enlarged from the actual object. The scale is usually given as a ratio (e.g., 1:100), meaning 1 unit on the diagram represents 100 units in reality.
Key facts to remember
- 1Perimeter is the distance around a 2D shape, measured in units of length.
- 2Area is the amount of surface a 2D shape covers, measured in square units.
- 3Volume is the amount of space a 3D object occupies, measured in cubic units.
- 4The value of π (pi) is approximately 3.14159 or 22/7, depending on the question's instruction.
- 51 metre (m) = 100 centimetres (cm).
- 61 m² = 10,000 cm².
- 71 m³ = 1,000,000 cm³.
- 81 litre = 1000 cm³.
Worked examples
Example 1
A triangle has sides of length 5 cm, 12 cm, and 13 cm. Its base is 12 cm and its perpendicular height is 5 cm. Calculate its perimeter and area.
Answer
Perimeter = 30 cm, Area = 30 cm²
Always ensure the height is perpendicular to the base used.
Example 2
A disc has a radius of 7 cm. Using π = 22/7, calculate its circumference and area.
Answer
Circumference = 44 cm, Area = 154 cm²
Pay attention to the value of π given in the question, if any.
Example 3
A shape is formed by a rectangle with length 10 cm and width 6 cm, with a semi-disc attached to one of its 6 cm sides. Calculate the total area of the shape. Use π = 3.14.
Answer
Total Area = 74.13 cm²
For perimeter of composite shapes, only add the outer edges. The internal line between the rectangle and semi-disc is not part of the perimeter.
Example 4
A rectangular solid has a length of 8 cm, a width of 5 cm, and a height of 10 cm. Calculate its volume.
Answer
Volume = 400 cm³
Remember that volume is measured in cubic units.
Example 5
A cylindrical can has a radius of 3 cm and a height of 10 cm. Calculate its volume, giving your answer in terms of π.
Answer
Volume = 90π cm³
If the question asks for the answer 'in terms of π', do not substitute a numerical value for π.
Example 6
A scaled diagram of a garden shows a path with a length of 4 cm. The scale of the diagram is 1:200. What is the actual length of the path in metres?
Answer
The actual length of the path is 8 m.
Always check if the question requires the answer in specific units (e.g., metres instead of centimetres).
Common mistakes
- ✗Confusing the formulas for perimeter and area, or circumference and area.
- ✗Using the diameter instead of the radius (or vice versa) in disc/cylinder formulas.
- ✗Forgetting to include the correct units (cm, cm², cm³) in the final answer.
- ✗Incorrectly identifying the base and perpendicular height for the area of a triangle.
- ✗Not dividing by 2 when calculating the area or circumference of a semi-disc.
- ✗Misinterpreting the scale in scaled diagrams (e.g., dividing instead of multiplying, or vice versa).
Exam tips
- ★Always draw a clear diagram for the problem if one isn't provided, or add labels to an existing one.
- ★Write down the correct formula before substituting any values into it.
- ★Show all your steps clearly and logically, as marks are awarded for working.
- ★Double-check your calculations, especially when using π or dealing with multiple steps.
- ★Read the question carefully to ensure you answer exactly what is asked, including units and level of precision (e.g., 'in terms of π' or 'to one decimal place').
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