Geometry Circles: Angles, Chords and Tangents Worksheet (PDF)

Geometry math worksheet: Circles: Angles, Chords and Tangents – Maths PDF

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This circle angles worksheet for Geometry covers circles: angles, chords and tangents (Common Core G-C.A.1, G-C.A.2, G-C.A.3, G-C.A.4), with a free PDF to print.

These 20 problems practice central angles, inscribed angles, tangents, chords and circles of triangles (Common Core G-C.A.1 to G-C.A.4). The first problems are quick one-step checks of the definitions and formulas from the lesson. The middle problems ask for diagrams, equations and careful reasoning in the style of a high school Geometry class. The last six problems are applications with wheels, ponds and coordinates of circles.

Geometry · Common Core State Standards: G-C.A.1, G-C.A.2, G-C.A.3, G-C.A.4

Circle angles worksheet: exercises 1 to 10

Exercise 1 – Central angle

A central angle of a circle intercepts an arc of \(72^{\circ}\). Find the measure of the central angle.

Exercise 2 – Inscribed angle

An inscribed angle is half of its intercepted arc: 110/2 = 55.

An inscribed angle intercepts an arc of \(110^{\circ}\). Find the measure of the inscribed angle.

Exercise 3 – Find the arc

An inscribed angle measures \(47^{\circ}\). Find the measure of the arc it intercepts.

Exercise 4 – Angle in a semicircle

An angle inscribed in a semicircle is a right angle.

AB is a diameter of the circle and C is on the circle. If \(m\angle A = 35^{\circ}\), find \(m\angle B\).

Exercise 5 – Same arc, same angle

Two inscribed angles intercept the same arc. Their measures are \((3x + 5)^{\circ}\) and \((5x – 15)^{\circ}\). Find x and the angle measure.

Exercise 6 – Central and inscribed

A central angle measures \((4x – 30)^{\circ}\) and an inscribed angle on the same arc measures \((x + 20)^{\circ}\). Find x and both angles.

Exercise 7 – Inscribed quadrilateral

Opposite angles of an inscribed quadrilateral are supplementary.

Quadrilateral ABCD is inscribed in a circle. \(m\angle A = 82^{\circ}\) and \(m\angle B = 95^{\circ}\). Find \(m\angle C\) and \(m\angle D\).

Exercise 8 – Inscribed quadrilateral algebra

In an inscribed quadrilateral, \(m\angle A = (3x + 7)^{\circ}\) and the opposite angle \(m\angle C = (5x – 27)^{\circ}\). Find x and both angles.

Exercise 9 – Tangent and distance

A tangent is perpendicular to the radius at the point of tangency.

A circle has radius 5. From an outside point P, a tangent segment PA has length 12. Find the distance OP from the center to P.

Exercise 10 – Equal tangent segments

Tangent segments PA and PB are drawn to a circle from point P. \(PA = 3x + 4\) and \(PB = 5x – 10\). Find x and PA.

Exercises 11 and up

Exercise 11 – Tangent and radius angle

PA is tangent to a circle with center O at A. \(m\angle AOP = 64^{\circ}\). Find \(m\angle APO\).

Exercise 12 – Chord from a central angle

A circle has radius 10. A chord subtends a central angle of \(60^{\circ}\). Find the length of the chord.

Exercise 13 – Distance to a chord

Chord AB = 24 in a circle of radius 13.

A circle has radius 13 and a chord of length 24. Find the distance d from the center to the chord.

Exercise 14 – Arcs add to 360

Four arcs of a circle measure 80°, 95°, 110° and \(x\). Find x.

Exercise 15 – Tangent-chord angle

A tangent and a chord meet at a point on a circle. The angle between them is half of the intercepted arc. Find the angle if the arc is \(130^{\circ}\).

Exercise 16 – Chords crossing inside

Two chords cross inside a circle. The angle formed measures half the sum of the arcs it and its vertical angle intercept. Find the angle if those arcs are \(70^{\circ}\) and \(50^{\circ}\).

Exercise 17 – Circles of a triangle

A right triangle has legs 6 and 8 and hypotenuse 10. Find the radius of its circumscribed circle (the hypotenuse is a diameter) and the radius of its inscribed circle, which is \(\frac{a + b – c}{2}\).

Exercise 18 – All circles are similar

Circle A has radius 3 and circle B has radius 8. Find the scale factor of the dilation that maps A to B, the ratio of their circumferences and the ratio of their areas.

Exercise 19 – Ferris wheel

A Ferris wheel has radius 20 feet and 8 cars equally spaced around the rim. Find the central angle between neighboring cars and the straight-line distance between neighboring cars to the nearest tenth.

Exercise 20 – Round pond

Path PA is tangent to the pond.

A walking path from point P touches a circular pond at A and is tangent to it. The pond has radius 15 feet and PA is 20 feet. How far is P from the center of the pond?

Key facts behind this circle angles worksheet

Key facts

Central angle = intercepted arc; inscribed angle = half the arc.
An angle inscribed in a semicircle measures 90 degrees.
Opposite angles of an inscribed quadrilateral are supplementary.
A tangent is perpendicular to the radius at the point of tangency.
Tangent segments from one outside point are congruent.

How to use this circle angles worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

Download the circle angles worksheet as a PDF with the button below and keep it with your notes.

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