Free · Year 3 of high school · 50 minutes · 15 questions · no penalty

Practice maths class test — Year 3: the circle.

15 questions drawn at random from a bank of 25, on the block of the programme your teacher is finishing: every attempt is a different test. No penalty for wrong answers, so answer even if you are not sure. At the end you receive by email the correct answer to every question and your weak topics — with an account you see them straight away here on the page. The step-by-step solution unlocks with your first lesson.

A class test is the written test an Italian high-school class sits after each block of the programme; this practice version is multiple choice.

Test in ItalianThe questions of this simulator are in Italian, as in the real exam.

Test prepared by Gaetano Livornese · Aerospace engineer, Politecnico di Milano · maths and physics tutor · updated 13 October 2026

Programme covered by the test

Centre and radius from the general form
Degenerate circles and the condition for existence
Circle through three points or with its centre on a line
Relative position of a line and a circle
Tangents from a point and tangents parallel to a line
Length of a chord
Relative position of two circles and the radical axis
Families of circles and base points

Two worked examples from the test

Question 1 · Circonferenza nel piano cartesiano (retta e circonferenza, fasci) · Circonferenza

Determina centro e raggio della circonferenza di equazione $x^{2} + y^{2} + 8 x - 2 y - 8 = 0$ .

A $C (4, - 1)$ , $r = 5$
B $C (- 4, 1)$ , $r = 5$ correct answer
C $C (- 4, 1)$ , $r = 25$
D $C (- 8, 2)$ , $r = 5$

Solution

Confronta con la forma generale $x^{2} + y^{2} + a x + b y + c = 0$ : qui $a = 8$ , $b = - 2$ , $c = - 8$ .
Centro: $C (- \frac{a}{2}, - \frac{b}{2}) = C (- 4, 1) .$
Raggio:

$r = \frac{a ^{2}}{4} + \frac{b ^{2}}{4} - c = 16 + 1 + 8 = 25 = 5$

Il segno è la trappola classica: le coordinate del centro sono le opposte delle metà di $a$ e $b$ , quindi con $a = 8$ l'ascissa è $- 4$ e con $b = - 2$ l'ordinata è $+ 1$ .

Question 2 · Circonferenza nel piano cartesiano (retta e circonferenza, fasci) · Circonferenza

Per quale valore di $k$ l'equazione $x^{2} + y^{2} - 6 x + 2 y + k = 0$ rappresenta una circonferenza di raggio $3$ ?

A $k = - 1$
B $k = 1$ correct answer
C $k = 7$
D $k = 19$

Solution

Qui $a = - 6$ e $b = 2$ , quindi $\frac{a ^{2}}{4} + \frac{b ^{2}}{4} = 9 + 1 = 10 .$
La condizione va imposta sul quadrato del raggio, non sul raggio:

$10 - k = r^{2} = 9 \Rightarrow k = 1$

Verifica: $x^{2} + y^{2} - 6 x + 2 y + 1 = 0$ ha centro $C (3, - 1)$ e $r = 9 + 1 - 1 = 3$ .

Chi uguaglia $10 - k$ direttamente a $3$ ottiene $k = 7$ , che dà invece $r = 3$ .

In the online practice test you will find 13 more questions like these; the correct answer to every question arrives by email.

Prefer paper? Download the test as a PDF (15 questions, 4 pages · 75 KB, ready to print). You receive the solutions by email after taking it online.

Download the PDF

What a maths test on the circle looks like

Centre and radius from the general form, the circle through three points or with its centre on a line, tangents from an external point, the length of a chord and the relative position of two circles.

The mistake that costs the grade

Getting the signs wrong going from x² + y² + ax + by + c = 0 to the centre (−a/2, −b/2). Then, not checking that a²/4 + b²/4 − c is positive: if it is zero or negative the equation does not represent a real circle and the exercise has to be discussed, not solved.

How to use this practice test

Take it once cold, without revising and with the timer on: it is there to show you where you stand, not to look good. Then look at your weak topics, revise only those — usually two or three, not the whole chapter — and retake it after a few days. The 15 questions are drawn from a bank of 25, so the second attempt is not the same test and the scores are genuinely comparable.

Is the score not enough?

See which topics lost you the most marks, then read how to handle test anxiety — or we look at your results together: the diagnostic is free.

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