Mensuration And Calculation
Overview
This is the topic where knowing which formula to recall is worth as much as using it. There is no formulae sheet in any 9260 paper: the front cover lists a calculator and mathematical instruments and nothing else. Appendix 6.1 of the specification names the formulae you must carry in your head, and several of them live here: the circumference and area of a circle, Pythagoras, the three trigonometric ratios, the sine rule, the cosine rule, the area of a triangle as half ab sin C, the area of a trapezium and the volume of a prism.
You will use standard units of measure and change freely between them, including the compound units of speed and density; apply formulae for the area of triangles, parallelograms and trapezia and for the volume of a solid with a constant cross sectional area; work with the circumference and area of a circle and with composite shapes; and use Pythagoras and the trigonometric ratios in right-angled triangles in two-dimensional figures. On the Extension Tier you add surface area and volume of spheres, pyramids, cones, composite solids and frustums, the relationships between lengths, areas and volumes in similar figures, arc lengths and sectors, trigonometry in three dimensional figures, and the sine and cosine rules.
Objectives
- [Core] use standard units of measure and related concepts (length, area, volume/capacity, mass, time, money etc); change freely between related standard units (eg time, length, area, volume/capacity, mass) and compound units (eg speed and density) (Notes: 24 and 12 hour clock for times)
- [Core] know and apply formulae to calculate: area of triangles, parallelograms, trapezia; volume of 3D shapes using V = Ah where A is the constant cross sectional area and h is the height/length
- [Core] know and use the formulae: circumference of a circle = 2πr = πd; area of a circle = πr²
- [Core] calculate perimeters and areas of 2D shapes, including composite shapes
- [Extension] surface area and volume of spheres, pyramids, cones and composite solids including composite shapes and frustums of pyramids and cones (Notes: solutions in terms of π may be asked for)
- [Extension] use the relationships between lengths, areas and volumes in similar figures
- [Extension] calculate arc lengths, angles and areas of sectors of circles
- [Core] know the formula for: Pythagoras' theorem, a² + b² = c², and the trigonometric ratios sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse and tan θ = opposite/adjacent
- [Core] apply them to find lengths and angles in right-angled triangles in two-dimensional figures
- [Extension] including 3D figures
- [Extension] know and apply the sine rule, a/sin A = b/sin B = c/sin C, and cosine rule, a² = b² + c² − 2bc cos A, to find unknown lengths and angles
- [Extension] know and apply Area = ½ ab sin C to calculate the area, sides or angles of any triangle
Mind map
This topic is mapped out so you can see how the ideas connect.
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Lesson Note
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Lesson Evaluation
Congratulations on completing the lesson on Mensuration And Calculation. Now that youve explored the key concepts and ideas, its time to put your knowledge to the test. This section offers a variety of practice questions designed to reinforce your understanding and help you gauge your grasp of the material.
You will encounter a mix of question types, including multiple-choice questions, short answer questions, and essay questions. Each question is thoughtfully crafted to assess different aspects of your knowledge and critical thinking skills.
Use this evaluation section as an opportunity to reinforce your understanding of the topic and to identify any areas where you may need additional study. Don't be discouraged by any challenges you encounter; instead, view them as opportunities for growth and improvement.
- How many square centimetres are there in one square metre? A. 100 B. 1000 C. 10 000 D. 1 000 000 Answer: C
- A right-angled triangle has a hypotenuse of 73 cm and one other side of 48 cm. What is the third side? A. 25 cm B. 55 cm C. 61 cm D. 87 cm Answer: B
- Two similar solids have corresponding lengths 10 cm and 12 cm. The smaller has volume 500 cm^3. What is the volume of the larger? A. 600 cm^3 B. 720 cm^3 C. 864 cm^3 D. 1000 cm^3 Answer: C
- Which formula will NOT be printed on a 9260 question paper when it is needed? A. The volume of a cone B. The volume of a sphere C. The area of a circle D. The curved surface area of a cone Answer: C
- A triangle has sides 16 cm, 9 cm and 20 cm. Which rule finds the angle opposite the 20 cm side? A. Pythagoras B. The sine rule C. The cosine rule D. Half ab sin C Answer: C
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