Entrance exams: geometry · 跟马丁学编程

Geometry starts with a sketch

The second pillar of entrance exams: geometry. Perimeters, areas, Pythagoras’ theorem, solids, angles. It sounds like a pile of formulae — but really it is mainly one skill: make a sketch.

A geometry question without a sketch is like a map read with your eyes shut. With a sketch the question mostly solves itself: you draw, you label known sizes, and suddenly you see what is missing and which way to go.

The formulae entrance exams really want are surprisingly few: perimeter and area of a square, a rectangle, a triangle and a circle, Pythagoras’ theorem, the sum of angles in a triangle (180°), the volume of a cuboid and a cylinder. That is the core. You have them in your head from earlier levels — today we dust them off in test mode.

And again, calm: geometry questions tend to be generous with marks and the method is marked. Even if the last step does not come out, a correct sketch and a started calculation carry marks. Pencil in hand, let’s draw.

Adam in a practice test skipped a question about a ladder against a wall — “too much text, I don’t get it”. Sofie shows it her way: she draws a wall, the ground, a slanted ladder. “See? A right-angled triangle. The ladder is the hypotenuse 5 metres, from the wall 3 metres — you look for the height.” Adam looks: “Ah, that is ordinary Pythagoras! 25 minus 9 is 16, height 4 metres.” Sofie shrugs: “The question was not hard. It was just dressed in text. A sketch undressed it.”

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A sketch does not have to be pretty or exact. It must have all the sizes from the question labelled and a question mark by what you look for. You draw a helper, not art.

A test method for a geometry question

Step 1: Sketch and labels. Read the question, draw the situation, by each line write a size from the question. By the sought quantity a question mark.

Step 2: 姓名 the shape. “That is a right-angled triangle.” “That is a rectangle with a circular hole.” Once you name the shape, the formula reports for duty on its own.

Step 3: Pick a formula according to the question. Perimeter = the path around (metres, cm). Area = the space inside (m², cm²). Volume = the space inside a solid (m³, cm³). The unit in the question gives away the formula.

Step 4: Substitute and calculate line by line. Pythagoras: c² = a² + b² (the hypotenuse last — it is the longest). Circle: circumference 2 · π · r, area π · r², with π ≈ 3.14. Watch diameter versus radius!

Step 5: Answer with a unit and check with sense. Hypotenuse longer than the legs? Area positive? The height of a house did not come out 250 m?

The most common test traps: a given diameter instead of a radius (divide by two), units in different systems (metres and centimetres in one question — unify), perimeter mixed up with area, and a “forgotten half” with a triangle (area = a · v / 2, not a · v).

Example 1: Pythagoras in text

Question (a test type): A ladder 5 m long leans against a wall. Its foot stands 3 m from the wall. To what height does the ladder reach?

Method: Sketch: wall vertical, ground horizontal, ladder slanted. A right angle between the wall and the ground.

Ladder = hypotenuse c = 5. Distance from the wall = leg a = 3. I look for leg b.

Pythagoras: b² = c² − a² = 25 − 9 = 16. So b = 4.

Sense check: 4 m is less than the length of the ladder 5 m — a leg shorter than the hypotenuse. It matches.

Answer: The ladder reaches a height of 4 m.

Example 2: the area of a composite shape

Question: A garden has the shape of a rectangle 12 × 8 m. In the middle is a circular pool with diameter 4 m. How many m² is left for lawn? (π ≈ 3.14)

Method: Sketch: a rectangle, a circle in it.

Rectangle: 12 · 8 = 96 m².

Circle: diameter 4 → radius 2. Area = 3.14 · 2² = 3.14 · 4 = 12.56 m².

Lawn = rectangle minus circle: 96 − 12.56 = 83.44 m².

Answer: 83.44 m² is left for lawn. The trap of the question was in the diameter — who put in 4 as the radius got a circle four times bigger.

Example 3: angles in a triangle

Question: In a triangle one angle is 90° and the second is four times bigger than the third. Find the other two angles.

Method: The sum of angles in a triangle is 180°. For the other two, 180 − 90 = 90° is left.

I label the third angle x, the second is 4x. Equation: x + 4x = 90, so 5x = 90, so x = 18°.

Second angle: 4 · 18 = 72°.

Check: 90 + 72 + 18 = 180°. It matches.

Answer: The other angles are 72° and 18°. Geometry and an equation in one question — a common mix in entrance exams.

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Diameter is not radius. The most profitable trap of test questions with a circle and a cylinder: the question gives a diameter and the formula wants a radius. Before you substitute, underline the word “diameter” or “radius” in the question and by a diameter at once write r = diameter/2. One underline, one saved mark.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. 开始 with ten. When it goes well, add more.

Pythagoras’ theorem — the most common geometry guest of entrance exams. Write a whole number.

Practise the questions

On paper

A test set. For every question a required sketch with labelled sizes — we train a habit, not speed.

A rectangular garden 15 × 9 m: calculate the perimeter (for a fence) and the area (for grass). Watch the units: m against m².
A right-angled triangle with legs 6 cm and 8 cm: calculate the hypotenuse and the area. (The area of a right-angled triangle = leg times leg divided by two.)
A circular flower bed with diameter 6 m: circumference and area, π ≈ 3.14. Radius first!
A cuboid aquarium 40 × 25 × 30 cm: how many litres of water does it hold? (1 litre = 1000 cm³.)

Now you 💪

Every question starts with a sketch with sizes and a question mark.
From a diameter you make a radius before you reach for a formula.
You answer with a unit and check the result with sense.

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Done when: You have four solved test questions, each with a sketch, the right unit and a sense check.

What to take from this lesson

A geometry question = a sketch + a named shape + a formula.
The unit in the question gives away the formula: m perimeter, m² area, m³ volume.
Pythagoras: the hypotenuse is the longest and is calculated from last.
Divide the diameter by two. Underline it in the question.
The method carries marks, even if the last step does not come out. Write it.