High school geometry fans, get ready to master circles! Our free Geometry Chapter 6.4 - 6.6 Quiz challenges your circle theorem quiz skills, arc length practice quiz abilities, and segment know-how. With instant feedback and personalized insights, you'll identify areas to improve and build your confidence before exams. This engaging geometry practice test helps you reinforce key theorems and formulas. Whether you're prepping for finals or looking for extra review, jump into a quick chords and arcs check and boost your confidence with the areas of circles and sectors quiz . Take the test, track your score, and dare to hit 100%! Dive in now to ace your high school geometry quiz.
What is the circumference of a circle with radius 7?
14?
7?
49?
28?
The circumference of a circle is calculated by C = 2?r. Substituting r = 7 gives C = 2? × 7 = 14?. This matches the first answer option exactly. Learn more about circle circumference.
What is the measure of an inscribed angle that intercepts an arc of 80°?
40°
80°
160°
20°
The inscribed angle theorem states that an inscribed angle is half the measure of its intercepted arc. Therefore, an inscribed angle intercepting an 80° arc measures 80°/2 = 40°. Any other choice contradicts this property. See inscribed angle theorem.
What is the length of an arc in a circle of radius 10 units subtended by a 60° central angle?
10?/3
5?/3
20?/3
60?
Arc length is found using L = r?, where ? is in radians. A 60° angle converts to ?/3 radians. Substituting r = 10 gives L = 10 × ?/3 = 10?/3. Options that do not simplify to this value are incorrect. Learn about arc length.
What is the area of a circle with diameter 8?
16?
8?
32?
64?
The area of a circle is given by A = ?r˛. Since the diameter is 8, the radius is 4. Thus, A = ? × 4˛ = 16?. None of the other areas match this calculation. Circle area formula.
What is the area of a sector with radius 6 and central angle of 120°?
12?
8?
18?
24?
The area of a sector is (?/360) × ?r˛. With r = 6 and ? = 120°, that is (120/360) × ? × 36. Simplifying gives (1/3) × 36? = 12?. Thus 12? is correct and matches the first answer option. Sector area explained.
A tangent and a secant are drawn from an external point. The secant intercepts the circle at points A and B such that the external segment is 4 and the internal segment is 6. What is the length of the tangent segment?
2?10
?10
4?10
10
The tangent-secant theorem states that the square of the tangent segment equals the product of the secant's external segment and its entire length. Here, the external secant is 4 and the internal part is 6, so the whole secant is 10. Thus, tangent˛ = 4 × 10 = 40, giving tangent length = ?40 = 2?10. This confirms 2?10 is the correct value. Tangent-secant theorem.
Two chords intersect inside a circle at point E. If AE = 2, EB = 3, and CE = 4, what is the length of ED?
1.5
0.75
2
6
When two chords intersect inside a circle, the products of the segment lengths of each chord are equal: AE × EB = CE × ED. Given AE = 2 and EB = 3, their product is 6. With CE = 4, we solve 4 × ED = 6 to find ED = 1.5. Thus the intersection property yields ED = 1.5. Chord intersection theorem.
What is the measure of the angle formed by two secants that intersect outside the circle if they intercept arcs of 50° and 130°?
40°
60°
80°
90°
The angle formed by two secants intersecting outside a circle is half the difference of the measures of the intercepted arcs. Here the arcs are 130° and 50°, so the difference is 80°. Half of 80° gives 40° for the angle outside the circle. Therefore, 40° is the correct answer. Secant-secant angle theorem.
In a circle of radius 13, a chord is located 5 units from the center. What is the length of the chord?
24
26
10
12
The length of a chord can be found using the formula 2?(r˛ ? d˛), where d is the distance from the center to the chord. Here r = 13 and d = 5, so r˛ ? d˛ = 169 ? 25 = 144. Taking the square root gives 12, and doubling yields 24. Hence, the chord length is 24 units. Circle chord length.
Two chords intersect inside a circle and intercept arcs of 100° and 80° respectively. What is the measure of the angle formed by the chords?
90°
50°
40°
180°
When two chords intersect inside a circle, the measure of the angle formed is half the sum of the measures of the intercepted arcs. Given intercepted arcs of 100° and 80°, their sum is 180°. Half of 180° is 90°, which is the angle measure. Therefore, the correct answer is 90°. Intersecting chords theorem.
What is the area of the segment in a circle with radius 10 and central angle 60°?
50?/3 ? 25?3
100?/6 ? 50?3
50?/3 + 25?3
100?/3 ? 25?3
The area of a circular segment is found by subtracting the area of the triangular portion from the area of the sector. For r = 10 and central angle 60°, the sector area is (60/360) × ? × 100 = 100?/6 = 50?/3. The triangle area is (1/2)r˛ sin ? = (1/2) × 100 × sin 60° = 50 × (?3/2) = 25?3. Subtracting gives segment area 50?/3 ? 25?3. Circle segment area.
A tangent touches a circle at point P, and chord PA intercepts an arc measuring 70°. What is the angle between the tangent and the chord at point P?
35°
70°
140°
55°
The angle between a tangent and a chord through the point of contact equals half the measure of the intercepted arc. Here the intercepted arc measures 70°, so the angle is 70°/2 = 35°. This theorem ensures that 35° is correct. No other option follows this property. Tangent-chord angle theorem.
A circle has an area of 36?. Two parallel tangents are drawn on opposite sides of the circle. What is the distance between these two tangents?
12
6
18
36
Parallel tangents to a circle are always a distance of twice the radius (the diameter) apart. The circles area is 36?, so ?r˛ = 36? gives r˛ = 36 and r = 6. Thus, the distance between the tangents is 2r = 12. Therefore, 12 is correct. Circle tangent properties.
The arc length of a circle is 15? for a central angle of 135°. What is the radius of the circle?
20
10
15
25
Arc length L = r?, with ? in radians. A central angle of 135° is 135° × (?/180°) = 3?/4 radians. Setting L = 15? gives 15? = r × 3?/4, so r = (15?)/(3?/4) = 15 × 4/3 = 20. Hence, the radius is 20 units. Arc length formula.
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Study Outcomes
Understand Circle Theorems -
Identify and explain key circle theorems, including inscribed angles, central angles, and angle relationships formed by chords, tangents, and secants.
Apply Arc Length Formulas -
Use the relationship between arc measure and circle circumference to calculate arc lengths accurately in various problem contexts.
Solve Segment Relationship Problems -
Determine lengths and measures of circle segments by applying properties of intersecting chords, secants, and tangent-secant angles.
Analyze Inscribed and Central Angles -
Distinguish between inscribed and central angles and compute their measures using theorems relating angles to intercepted arcs.
Calculate Areas and Perimeters of Segments -
Compute the area and perimeter of circular segments by integrating arc length and triangle or sector formulas for complete solutions.
Improve Timed Quiz Performance -
Develop speed and accuracy under timed conditions to boost confidence and readiness for high-school geometry assessments.
Cheat Sheet
Central and Inscribed Angle Theorems -
Central angles subtend arcs equal to their measure, while inscribed angles measure half the intercepted arc, a key circle theorem quiz concept. For example, an inscribed 50° angle always spans a 100° arc, using the "half-step" mnemonic to lock it in. (Source: MIT OpenCourseWare)
Arc Length and Sector Area Formulas -
When prepping for your Geometry Chapter 6.4 - 6.6 Quiz, remember arc length = r¡θ (θ in radians) and sector area = ½r²θ. For instance, a 5 cm radius circle with a 2 rad arc has length 10 cm and sector area 25 cm². (Source: Khan Academy)
Angles from Two Intersecting Chords -
An interior angle formed by intersecting chords equals half the sum of the intercepted arcs: â = ½(arcâ + arcâ). In a circle theorem quiz problem, chords AB and CD crossing at E give â AEC = ½(arc AC + arc BD). (Source: University of Texas Math)
Angles from Secants and Tangents Outside -
Angles with vertices outside the circle follow â = ½(outer arc - inner arc), a staple in arc length practice quizzes. For example, two secants from P yield â P = ½(arc BD - arc AC). (Source: Purplemath)
Power of a Point and Segment Relationships -
Chord - chord, secant - secant, and tangent - secant segments satisfy PA¡PB = PC¡PD or tangent² = external¡whole, a powerful fact for geometry practice tests. Use this to quickly find missing lengths when two chords or secants intersect. (Source: Paul's Online Math Notes)
