Question 1
A circle has radius 4.
The area of the circle is kπ. What is k?
- 16
- 26
- 64
- 8
- 4
Show the worked solution
- Area = πr² = π × 4² = 16π, so k = 16.
- Answer: 16.
Geometry (text only)
Geometry questions you can solve without a diagram
These geometry questions are stated entirely in words, with no diagram to read. Each one rests on a small set of facts: angles in a triangle sum to 180 degrees, the Pythagorean theorem relates the sides of a right triangle, circle area and circumference follow from the radius, and the distance between two points is the hypotenuse of the gap in each coordinate.
Working without a figure sounds harder and usually is not, because the words state exactly what a diagram would show and nothing is left to measurement. What it does require is recognising the standard triples on sight — 3-4-5, 5-12-13 and 8-15-17 — since a question giving a leg of 8 and a hypotenuse of 10 is answered by recognition rather than by squaring anything.
Try it against the clock. Geometry (text only) runs at 90s easy, 105s medium and 120s hard. The clock is stamped and judged on our server, so the limit you see is the deadline that is actually enforced — and it starts when you tap Start, not while the question is loading.
How to answer text-only geometry questions from the stated facts
-
Find the third angle by subtraction
Angles in a triangle total 180 degrees, so two given angles fix the third. This is the most frequent single question in the set and needs no construction at all.
-
Recognise the Pythagorean triple before calculating
A leg of 8 with a hypotenuse of 10 is the 3-4-5 triple doubled, so the missing leg is 6. Squaring and subtracting gets there too, and takes several times as long.
-
Read coordinate distance as a right triangle
From (8, −2) to (11, −2) the horizontal gap is 3 and the vertical gap is 0, so the distance is 3. When one gap is zero the distance is simply the other one.
-
Work circles from the radius outward
Area is pi times the radius squared and circumference is two pi times the radius. Check which one the question wants, since both values are always on the option list.
The facts these questions are built from
| Given | Use |
|---|---|
| Two angles of a triangle | Third = 180 − the other two |
| Leg 8, hypotenuse 10 | 3-4-5 doubled, missing leg 6 |
| Legs 8 and 15 | 8-15-17, hypotenuse 17 |
| Two points on a line | Distance = √(Δx² + Δy²) |
| Radius r | Area πr², circumference 2πr |
3 sample geometry questions, with worked solutions
Generated by the same engine that mints the Quant Daily daily. Each answer is computed from the numbers printed in the question, and the walkthrough below each one is the engine's own working — not a solution written afterwards. Reload this page's live drill and you get different numbers.
Question 2
Point P is (1, 1) and point Q is (4, 5).
What is the distance between the two points?
- 3
- 6
- 4
- 5
- 7
Show the worked solution
- Δx = 3, Δy = 4.
- Distance = √(3² + 4²) = 5.
- Answer: 5.
Question 3
A right triangle has legs 5 and 12.
What is the perimeter?
- 18
- 17
- 60
- 13
- 30
Show the worked solution
- Hypotenuse = 13.
- Perimeter = 5 + 12 + 13 = 30.
- Answer: 30.
Common mistakes this drill is built from
The wrong options are not random numbers. Each one is the result of a specific careless error for this topic, so picking one tells you which habit is costing you marks.
Treating a leg as the hypotenuse
The hypotenuse is always the longest side and always opposite the right angle. Swapping it turns a subtraction into an addition and gives a larger wrong answer.
Confusing area with circumference
Both follow from the radius and both are printed. Underline which one the question asks for before computing.
Adding the coordinate gaps instead of combining them
Distance is the square root of the sum of the squares, not the sum of the gaps. The two agree only when one of the gaps is zero.
Why your answer came out wrong
The specific wrong numbers this topic produces, and what each one tells you about the step you took. If you have just got a question wrong and want to know which habit did it, start here.
A right triangle has one leg 8 and hypotenuse 10 — is the missing leg 12.8?
It is 6. Getting a value larger than the hypotenuse means the 10 was treated as a leg and the two sides were added under the root instead of subtracted. The hypotenuse is the longest side, so any leg you find must come out smaller than it.
For points (8, −2) and (11, −2), is the distance √(3² + 2²)?
The y coordinates are both −2, so the vertical gap is 0, not 2. The distance is simply 3. Using the coordinate value rather than the difference between the coordinates is a frequent slip, especially when the numbers are negative.
A circle has radius 7 — is the area 44?
44 is roughly the circumference, which is 2πr. The area is πr², which is about 154. Both are derived from the radius and both appear among the options, so the deciding step is reading which quantity the question actually asked for.
FAQ
Do these questions include diagrams?
No — every question is stated in words, so there is nothing to interpret visually. The text gives you exactly the measurements a figure would carry, which makes the questions well suited to a quick timed drill on a phone.
Which Pythagorean triples should I know on sight?
3-4-5, 5-12-13 and 8-15-17, together with their multiples such as 6-8-10 and 9-12-15. Recognising one turns a two-step calculation into an immediate answer, and exam questions are built from these far more often than from arbitrary numbers.
How do I find the distance between two points quickly?
Take the difference in the x values and the difference in the y values, then combine them as the legs of a right triangle. If either difference is zero the distance is just the other one, and if the pair forms a known triple you can read the answer off directly.
Practise the next topic
- All 51 drill types — the shared daily cycles 36 of 51, with every published time limit.
- Mensuration questions — the formulas — mensuration questions, timed.
- Exponent rules practice — evaluate powers and roots on sight — exponent rules practice, timed.
- Speed tricks — the seven mental-math shortcuts that make a 15-second answer possible.
- How to improve calculation speed — the six-week arc these drills fit into.
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