Geometry, Measures and Constructions Depth
Geometry, Measures and Constructions Depth combines the harder end of Geometry and Measures with Perimeter, Area, Surface Area and Volume into multi-step problems, including compound shapes, surface area and volume of solids, bearings and ruler-and-compass constructions such as bisectors and loci.
Before you start
Make sure you're comfortable with these topics first:
Method
- Always sketch or annotate the diagram given, or draw your own if none is provided, marking on every length and angle from the question as you read it.
- For compound-shape area or perimeter questions, split the shape into simple rectangles, triangles, or circles and sectors, working out any missing lengths first.
- For angle problems, build up from known angle facts one structured step at a time, stating which rule justifies each new angle, such as angles on a straight line, angles in a triangle, or angles between parallel lines.
- For constructions, keep compass arcs and construction lines visible on the diagram rather than rubbing them out, since both accuracy and correct method are assessed, and set the compass precisely to the required radius.
- For loci and scale drawings, convert every real-world distance in the question to the given scale before drawing or measuring, and label the scale used.
- For surface area, work face by face, or curved surface plus circular ends for a cylinder, and check no face has been counted twice or missed, especially for an open shape such as a tank with no lid.
- Give the correct units in every answer, checking whether the question wants a length, an area or a volume, since the wrong units lose the accuracy mark even when the number is correct.
Worked example
A cylindrical candle has a radius of 4 cm and a height of 10 cm. It is packed in a box shaped like a cuboid with a square base, so that the candle exactly touches all four sides of the box, and the height of the box exactly equals the height of the candle. Work out the volume of empty space inside the box once the candle is packed inside, giving your answer to 3 significant figures. (Use pi = 3.14159.)
- Find the side length of the box's square base: since the candle touches all four sides, this equals the candle's diameter, 2 x 4 = 8 cm.
- Find the volume of the box: 8 x 8 x 10 = 640 cm^3.
- Find the volume of the cylindrical candle: pi x r^2 x h = pi x 4^2 x 10 = 160 x pi.
- Evaluate using pi = 3.14159: 160 x 3.14159 = 502.6544 cm^3.
- Subtract to find the empty space: 640 - 502.6544 = 137.3456, which rounds to 137 cm^3 (3 s.f.).
Practice questions
Type your answer and press Check to be marked straight away, or reveal the answer and mark yourself.
Q1A shape is made from a rectangle measuring 12 cm by 8 cm, with a right-angled triangle of base 6 cm and height 8 cm removed from one corner. Work out the area of the remaining shape.Show answer
Answer: 72 cm^2 (rectangle area = 96 cm^2, triangle area = 0.5 x 6 x 8 = 24 cm^2, remaining = 96 - 24 = 72 cm^2)
Q2A circle has a radius of 10 cm. Work out the area of a sector with an angle of 72 degrees at the centre, giving your answer to 3 significant figures. (Use pi = 3.14159.)Show answer
Answer: 62.8 cm^2 (full circle area = pi x 100 = 314.159, sector = 314.159 x 72/360 = 62.8318, rounds to 62.8)
Q3Work out the size of each interior angle of a regular decagon (10 sides).Show answer
Answer: 144 degrees (angle sum = (10 - 2) x 180 = 1440, each angle = 1440 / 10 = 144)
Q4Two parallel lines are cut by a transversal. One of the angles formed is 63 degrees. State the size of the co-interior (allied) angle on the same side of the transversal, and explain why.Show answer
Answer: 117 degrees, because co-interior angles between parallel lines always sum to 180 degrees (180 - 63 = 117)
Q5An open-top fish tank, with no lid, is a cuboid with length 50 cm, width 30 cm and height 25 cm. Work out the total area of glass needed to make the tank.Show answer
Answer: 5500 cm^2 (base 50 x 30 = 1500, two faces of 50 x 25 = 2500, two faces of 30 x 25 = 1500, total = 1500 + 2500 + 1500 = 5500, with no top face since the tank is open)
Q6On a map with a scale of 1 : 25,000, two villages are 6.8 cm apart. Work out the real-life distance between the villages, in kilometres.Show answer
Answer: 1.7 km (6.8 x 25,000 = 170,000 cm = 1700 m = 1.7 km)
Q7Describe the locus of points that are equidistant from two fixed points, A and B.Show answer
Answer: The perpendicular bisector of the line segment AB
Exam-style questions
Written in the style of a 13+ Common Entrance exam paper, with a full mark scheme.
A running track is made from a rectangle with a semicircle attached to each of the two shorter ends. The rectangular section measures 80 m by 60 m, and each semicircle has a diameter equal to the 60 m width. (a) Work out the perimeter of the track, giving your answer to 3 significant figures. (b) Work out the area enclosed by the track, giving your answer to 3 significant figures. (Use pi = 3.14159.)
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A ship sails from port P on a bearing of 062 degrees for 40 km to point Q. (a) Work out the bearing of P from Q (the back bearing). (b) The ship then sails from Q on a bearing of 152 degrees for 30 km to point R. The angle PQR formed at Q, between the ship's incoming and outgoing paths, is a right angle. Work out the direct distance from P to R.
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Free printable worksheet
Want more practice on paper? Download the geometry, measures and constructions depth worksheet pack - 16 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 19 of 13+ Maths Workbook, the whole course as one free printable PDF.
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