About Civil Engineering
Civil engineering questions ask about the things that get measured on site: the area of a plot, the capacity of a tank, the materials for a batch of concrete, the weight of a steel bar, the gradient of a road and the bearing of a survey line. Each is a short piece of arithmetic once the shape or the ratio is recognised.
What you need to understand
- A rectangle is length times breadth. A triangle is half base times height. A trapezium - a plot with two parallel sides - is the mean of the two parallel sides multiplied by the distance between them.
- A circle has area pi r squared, or pi d squared over four. Working from the diameter is where the factor of four is most often lost.
- A cylinder holds the area of its base multiplied by its depth, and its base is a circle. A cuboid holds length times breadth times depth.
- Concrete mixes are given as parts by volume, cement : sand : aggregate. In a 1 : 2 : 4 mix the sand is twice the cement and the aggregate is four times it, whatever the size of the batch.
- A cubic metre of concrete is not a cubic metre of dry materials, because the sand and aggregate fill the gaps between each other and the water takes up space. A question that allows for this states the factor, so use the figure you are given.
- A round steel bar weighs the diameter squared over 162 kilograms for each metre of length, with the diameter in millimetres. It is a site shortcut worth memorising.
- A gradient is the rise divided by the horizontal distance, and as a percentage that is multiplied by 100. A gradient of 1 in 50 is the same as 2 per cent.
- A whole circle bearing is measured clockwise from north, from 0 to 360 degrees. Surveying in quadrants measures from north or south towards east or west instead.
- Converting a quadrant bearing: north-east is the angle itself, south-east is 180 minus it, south-west is 180 plus it, and north-west is 360 minus it.
- Density is mass over volume, and the unit weight of a soil or a concrete follows from it once the water content is known.
How to work through these questions
- Name the shape before reaching for a formula. Most errors in this topic come from using the rectangle formula on something that is not a rectangle.
- Check the units of length before multiplying them together. Areas come out in square units and volumes in cubic units, and both are easy to get a factor of a thousand wrong on.
- For a mix, work out the parts from the cement. If the ratio is 1 : 2 : 4, one measure of cement brings two of sand and four of aggregate with it.
- For a steel bar, square the diameter first and then apply the formula. Doing it the other way round loses the whole point of the relation.
- For a gradient, decide whether the question wants the ratio, the percentage or the angle, and check the answer is of a sensible size for a road - a main road is rarely steeper than about 6 per cent.
- For bearings, sketch the compass points and mark the line. A picture settles the quadrant conversion far faster than remembering which sign goes where.
Mistakes that cost marks
- Using only one of the parallel sides of a trapezium instead of their mean. The area lies between the two, not at either of them.
- Using the diameter where the radius belongs in pi r squared, which makes the answer four times too large.
- Reading a mix ratio the wrong way round and quoting the sand where the aggregate was asked for.
- Forgetting the factor of four that turns a diameter into a radius when working out the volume of a tank.
- Quoting the weight of a single metre of bar as the answer when the whole length was asked for.
- Turning a gradient upside down. The rise goes on top and the run underneath, and the answer must be small for a road.
- Getting the quadrant conversion backwards - adding 180 where it should be subtracted, so the line points into the wrong quarter of the compass.
Worked example
A trapezoidal plot has parallel sides of 40 m and 60 m, and the distance between them is 25 m. What is its area, and what would a road with a rise of 5 m over a distance of 200 m have as its gradient?
- For the plot, start with the mean of the parallel sides: (40 + 60) / 2 = 50 m.
- Multiply by the distance between them: 50 x 25 = 1250 square metres.
- Check it against the rectangle it sits inside. A 60 by 25 rectangle is 1500 square metres, and a trapezium between 40 and 60 m wide must be smaller than that and larger than 40 x 25 = 1000. It is.
- For the road, the gradient is the rise over the horizontal distance: 5 / 200 = 0.025.
- As a percentage that is 2.5 per cent, a slope of 1 in 40 - steep but possible, so the answer is of a sensible size.
Answer: Area 1250 square metres; gradient 2.5 per cent, or 1 in 40
Practice questions with answers
A few Civil Engineering questions with the full solution shown, so you can see
how the method is applied before you attempt the timed set.
Question 1
A plot of land is a trapezium whose parallel sides are 38 m and 98 m and whose height is 16 m. What is its area, in square metres?
-
A
2176
-
B
1862
-
C
84
-
D
1088
Answer: Option D — with explanation
A trapezium's area is the mean of its two parallel sides multiplied by the distance between them: A = (a + b) / 2 x h.
The mean of 38 and 98 is 68 m, and 68 x 16 = 1088 square metres. Multiplying the two sides together and halving, or using only one of them, gives the other numbers offered - both of which describe a different shape.
Common mistakes
- 2176 is not the answer: 2176
- 1862 is not the answer: 1862
- 84 is not the answer: 84
Question 2
A cylindrical water tank has a diameter of 10 m and a depth of 2 m. What is its capacity, in cubic metres? Give the answer to two decimal places.
-
A
628.32
-
B
157.08
-
C
62.83
-
D
200
Answer: Option B — with explanation
The volume of a cylinder is the area of its circular base times the depth, and the base area is pi times the radius squared - or pi d squared over four if the diameter is what you have.
3.1416 x 10^2 / 4 = 78.54 square metres of base, and multiplying by a depth of 2 m gives 157.08 cubic metres. Using the diameter in place of the radius somewhere in the working is the mistake that produces the other answers.
Common mistakes
- 200 is not the answer: 200
- 62.83 is not the answer: 62.83
- 628.32 is not the answer: 628.32
Question 3
In a concrete mix of 1 : 2 : 4 by volume, 3 cubic metres of cement are used. How much coarse aggregate is needed, in cubic metres?
Answer: Option C — with explanation
In a 1 : 2 : 4 mix the numbers are parts by volume, so the aggregate is four times the cement and the sand is twice it.
With 3 cubic metres of cement the aggregate is 4 x 3 = 12 cubic metres, and the sand is 2 x 3 = 6 cubic metres, making 21 cubic metres of materials in all. Giving the sand instead of the aggregate is the commonest slip with a three-part ratio.
Common mistakes
- 9 is not the answer: 9
- 3 is not the answer: 3
- 6 is not the answer: 6
Question 4
What does a steel bar of diameter 32 mm weigh if it is 4 m long? Take its weight as d^2 / 162 kilograms per metre, and give the answer in kilograms.
-
A
6.32
-
B
25.28
-
C
50.57
-
D
0.79
Answer: Option B — with explanation
The usual site formula for a round steel bar is d squared over 162 kilograms for each metre of length, with the diameter in millimetres.
32^2 / 162 = 6.32 kg per metre, and over 4 m that is 25.28 kg. Using the diameter rather than its square, or quoting the weight of a single metre, gives the other numbers offered.
Common mistakes
- 50.57 is not the answer: 50.57
- 6.32 is not the answer: 6.32
- 0.79 is not the answer: 0.79
Question 5
A road rises 82 m over a horizontal distance of 400 m. What is its gradient, as a percentage?
-
A
41/2
-
B
41/200
-
C
20000/41
-
D
205
Answer: Option A — with explanation
A gradient compares the rise with the horizontal distance, and as a percentage it is the rise divided by the run, multiplied by 100.
82 / 400 x 100 = 41/2 per cent, which is the same as a slope of 1 in 200/41. Turning the fraction upside down - the run over the rise - gives 20000/41, which is the slope described the other way round.
Common mistakes
- 41/200 is not the answer: 41/200
- 20000/41 is not the answer: 20000/41
- 205 is not the answer: 205
Frequently asked questions
How do I remember the formula for a trapezium?
It is the width the plot would have if the sloping sides were squared off - the mean of the two parallel sides - multiplied by the distance between them. If both sides were the same length it would be a rectangle, and the formula gives exactly that.
Why is the weight of a steel bar given as d squared over 162?
It comes from the volume of a round bar times the density of steel, tidied up so that the diameter in millimetres gives kilograms per metre directly. It is a shortcut for the site, not a law of nature, which is why questions usually quote it.
What does a 1 : 2 : 4 mix mean in practice?
One part of cement, two parts of sand and four parts of aggregate, measured by volume. It is a general-purpose mix. A richer mix has less aggregate in proportion and a leaner one has more.
When is a bearing measured the other way round?
Only when the question says so. Whole circle bearings are always clockwise from north; quadrant bearings start at north or south and turn towards east or west. Read the letters before doing any arithmetic.
Do I need to allow for the water in a concrete mix?
Only if the question gives you the factor. A mix of 1 : 2 : 4 by volume does not yield that volume of concrete, because of the water and the gaps between the particles, but the allowance depends on the mix and the question has to state it.
