Circles With and Without Coordinates
Unit 5 of Geometry. 14 questions below, each with the working. Every answer was checked by a second pass before it was published.
Similarity of circles, inscribed angles, radii and chords, tangents, arc length and sectors, areas, equations of circles.
How this unit is tested
Start by sorting every circle problem into one of two worlds: pure geometric relationships (angles, chords, tangents, arcs) or coordinate relationships (equations, centers, radii on the plane). For the geometric world, always identify first whether the angle in question is central or inscribed, since the Inscribed Angle Theorem (half the arc) is the single most tested fact and the most common source of factor-of-2 errors. Draw the radius to any point of tangency immediately — a tangent line is perpendicular to the radius at that point, and that right angle is almost always the key to setting up a Pythagorean relationship or solving for a missing length.
For chords, remember that a radius perpendicular to a chord bisects it; this converts a chord problem into a right-triangle problem with the radius as hypotenuse. For arc length and sector area, treat the central angle as a fraction of the full circle (angle/360) and multiply that fraction by the full-circle quantity (circumference $2\pi r$ for arc length, area $\pi r^2$ for sector area) — do not mix up which formula uses which fraction.
For coordinate geometry, the equation of a circle comes directly from the distance formula: any point $(x,y)$ on the circle is a fixed distance $r$ from the center $(h,k)$, giving $(x-h)^2+(y-k)^2=r^2$. When given a general (expanded) equation, complete the square on the $x$-terms and $y$-terms separately to recover center and radius. When given two points (such as endpoints of a diameter), use the midpoint formula for the center and the distance formula for the radius.
Finally, remember similarity: every circle is similar to every other circle, because a translation of centers followed by a dilation by the ratio of radii always maps one circle exactly onto the other. This is a proof idea tested conceptually, not just a formula.
What you have to know
Similarity of Circles
Any two circles are similar. A translation mapping one center to the other, followed by a dilation with scale factor equal to the ratio of the radii, maps one circle exactly onto the other.
Inscribed Angle Theorem
The measure of an inscribed angle is half the measure of its intercepted arc (equivalently, half the central angle that intercepts the same arc).
Inscribed Angle in a Semicircle
An inscribed angle that intercepts a diameter (a semicircular arc) always measures 90 degrees.
Radius–Chord and Radius–Tangent Relationships
A radius drawn perpendicular to a chord bisects that chord. A radius drawn to a point of tangency is perpendicular to the tangent line at that point. Two tangent segments drawn from the same external point to a circle are congruent.
Arc Length and Sector Area
For a central angle of measure $\theta$ degrees on a circle of radius $r$: arc length $=\dfrac{\theta}{360}\cdot 2\pi r$, and sector area $=\dfrac{\theta}{360}\cdot \pi r^2$.
Equation of a Circle
A circle with center $(h,k)$ and radius $r$ is the set of points satisfying $(x-h)^2+(y-k)^2=r^2$. Expanded equations are converted to this form by completing the square.
14 practice questions
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Explain, using transformations, why every circle is similar to every other circle.
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Answer. Translate the center of one circle onto the center of the other, then dilate by a scale factor equal to the ratio of the two radii; this maps one circle exactly onto the other, proving they are similar.
Similarity requires a sequence of rigid motions and a dilation mapping one figure onto the other. A translation aligns centers, and since a circle is defined entirely by its center and radius, a single dilation by radius₂/radius₁ stretches or shrinks it to match exactly. -
A central angle intercepts an arc of 84°. What is the measure of an inscribed angle that intercepts the same arc?
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Answer. 42°
By the Inscribed Angle Theorem, an inscribed angle is half the measure of its intercepted arc. Half of 84° is 42°. -
Triangle ABC is inscribed in a circle with AC as a diameter. What is the measure of angle B?
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Answer. 90°
Since AC is a diameter, it subtends a semicircular arc of 180°. Angle B is an inscribed angle intercepting that arc, so its measure is half of 180°, which is 90°. -
Two inscribed angles, ∠P and ∠Q, both intercept the same arc of a circle. If ∠P = 3x + 5 and ∠Q = 5x − 15, find x.
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Answer. x = 10
Inscribed angles that intercept the same arc are congruent, so 3x+5 = 5x−15. Solving: 20 = 2x, so x = 10. -
A chord of a circle is 24 units long, and the circle's radius is 13 units. What is the distance from the center of the circle to the chord?
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Answer. 5 units
A radius perpendicular to a chord bisects it, so half the chord is 12. This forms a right triangle with hypotenuse 13 (radius) and leg 12 (half-chord); the other leg is $\sqrt{13^2-12^2}=\sqrt{25}=5$. -
From an external point P, two tangent segments are drawn to a circle, with lengths 3x − 4 and x + 10. Find the length of each tangent segment.
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Answer. 17 units each
Tangent segments from the same external point are congruent, so 3x−4 = x+10, giving 2x=14, x=7. Substituting, each length is 3(7)−4=17. -
A tangent segment from external point P to a circle of radius 9 has length 12. What is the distance from P to the center of the circle?
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Answer. 15 units
The radius drawn to the point of tangency is perpendicular to the tangent, forming a right triangle with legs 9 (radius) and 12 (tangent) and hypotenuse equal to the distance from P to the center: $\sqrt{9^2+12^2}=\sqrt{81+144}=\sqrt{225}=15$. -
A circle has radius 9. Find the exact arc length of an arc with a central angle of 120°.
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Answer. $6\pi$
Arc length equals the fraction of the circle times the circumference: $\frac{120}{360}\cdot 2\pi(9)=\frac{1}{3}\cdot 18\pi=6\pi$. -
A circle has radius 10. Find the exact area of a sector with a central angle of 72°.
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Answer. $20\pi$
Sector area equals the fraction of the circle times the total area: $\frac{72}{360}\cdot \pi(10)^2=\frac{1}{5}\cdot 100\pi=20\pi$. -
Write the equation of a circle with center (−2, 5) and radius 6.
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Answer. $(x+2)^2+(y-5)^2=36$
Substitute h=−2, k=5, r=6 into the standard form $(x-h)^2+(y-k)^2=r^2$: since $x-(-2)=x+2$, the equation is $(x+2)^2+(y-5)^2=36$. -
Find the center and radius of the circle given by $x^2+y^2-6x+4y-12=0$.
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Answer. Center (3, −2), radius 5
Group and complete the square: $(x^2-6x+9)+(y^2+4y+4)=12+9+4$, giving $(x-3)^2+(y+2)^2=25$. So the center is (3,−2) and radius is $\sqrt{25}=5$. -
A circle has a diameter with endpoints (2, 3) and (8, −5). Write the equation of the circle.
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Answer. $(x-5)^2+(y+1)^2=25$
The center is the midpoint: $\left(\frac{2+8}{2},\frac{3+(-5)}{2}\right)=(5,-1)$. The radius is the distance from the center to one endpoint: $\sqrt{(5-2)^2+(-1-3)^2}=\sqrt{9+16}=5$. So the equation is $(x-5)^2+(y+1)^2=25$. -
In a circle, two chords are each 16 units long. If one chord is 6 units from the center, how far from the center is the other chord?
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Answer. 6 units
Chords of equal length in the same circle are equidistant from the center. Since both chords have the same length (16), they must be the same distance (6 units) from the center. -
A circle has radius 4 and center (0,0). A larger, similar circle has radius 10 and the same center. What scale factor maps the smaller circle onto the larger one?
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Answer. 2.5
Since the centers already coincide, only a dilation is needed. The scale factor is the ratio of the radii: $\frac{10}{4}=2.5$.
What people get wrong
- Using the central angle directly as the inscribed angle (or vice versa) instead of applying the factor of 2 — always ask whether the vertex of the angle is at the center or on the circle before using its measure.
- Confusing the arc length formula with the sector area formula, since both start with angle/360 — remember arc length pairs with circumference ($2\pi r$, a length) and sector area pairs with area ($\pi r^2$, a squared quantity).
- Making a sign error when completing the square, writing $(x+h)^2$ instead of $(x-h)^2$ — after completing the square, always rewrite the binomial so the sign matches subtraction of the center coordinate, e.g. $x^2-6x$ becomes $(x-3)^2-9$, giving center x-coordinate +3, not −3.
- Treating the radius drawn to a point of tangency as anything other than perpendicular to the tangent — draw this radius explicitly and mark the right angle before setting up any Pythagorean or trigonometric relationship.
- Forgetting that a radius must be drawn perpendicular to a chord (not any radius) to guarantee it bisects the chord — the perpendicularity is the condition that must be checked or given.
- Assuming two chords are congruent just because they look similar in a diagram — congruence of chords follows only from equal distance from the center or equal arcs, which must be established, not assumed.
Drill this unit until it sticks
These questions come back on a schedule built from what you get wrong, alongside the rest of Geometry. Free, and no account needed to start.