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Introduction to Geometry / LEVEL 3 · DIFFICULTY 3/5

Polygons and Circles

Reason about turns, arcs and intersecting chords.

3 stages · 24 practice problems · two 6-question assessment forms

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MINI QUEST GEO 3.1Polygon Angle SumsRead the lesson
MINI QUEST GEO 3.2Arcs and Central AnglesRead the lesson
MINI QUEST GEO 3.3Intersecting ChordsRead the lesson
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STAGE GEO 3.1 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Polygon Angle Sums

Useful preparation: Right-Triangle Distance

Goal: Understand and apply polygon angle sums.

Before you begin: Right-Triangle Distance

Understand the idea

Drawing diagonals from one vertex divides a convex n-sided polygon into $\displaystyle n - 2$ triangles. Adding their angle sums counts each polygon interior angle exactly once.

interior sum=

\displaystyle \left(n - 2\right) \cdot 180^{\circ}

Choose and carry out a method

Count the sides, subtract two, and multiply by $\displaystyle 180^{\circ}$ . Regularity is unnecessary for the total but is needed to divide equally among vertices.

Check the reasoning

A triangle gives $\displaystyle 180^{\circ}$ and a quadrilateral gives $\displaystyle 360^{\circ}$ . Each extra side adds one triangle to the construction.

WORKED EXAMPLE 1

Find the sum of the interior angles of a convex 6-gon, in degrees.

Draw diagonals from one vertex to divide a convex polygon into triangles.
A 6-gon contains 4 triangles in this division.
The angle sum is $\displaystyle 4 \cdot 180 = 720^{\circ}$ .

720

WORKED EXAMPLE 2

Find the sum of the interior angles of a convex 7-gon, in degrees.

Draw diagonals from one vertex to divide a convex polygon into triangles.
A 7-gon contains 5 triangles in this division.
The angle sum is $\displaystyle 5 \cdot 180 = 900^{\circ}$ .

900

Common pitfalls

Possible mix-up: There are n triangles in the construction.

A fan from one vertex creates $\displaystyle n - 2$ triangles.

Possible mix-up: A correct numerical answer alone explains the method.

State the governing relationship and check the conditions described above.

Explain it to yourself

Why does a polygon’s total not determine each separate angle?

Preview the eight practice prompts

Problem 1
Find the sum of the interior angles of a convex 9-gon, in degrees.
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Problem 2
Find the sum of the interior angles of a convex 10-gon, in degrees.
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Problem 3
Find the sum of the interior angles of a convex 11-gon, in degrees.
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Problem 4
Find the sum of the interior angles of a convex 12-gon, in degrees.
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Problem 5
Find the sum of the interior angles of a convex 13-gon, in degrees.
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Problem 6
Find the sum of the interior angles of a convex 14-gon, in degrees.
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Problem 7New context
A convex garden border has 15 straight sides. What is the sum of its interior corner angles in degrees?
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Problem 8New context
A convex garden border has 16 straight sides. What is the sum of its interior corner angles in degrees?
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Open stage GEO 3.1 in the student workspace →

STAGE GEO 3.2 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Arcs and Central Angles

Useful preparation: Polygon Angle Sums

Goal: Understand and apply arcs and central angles.

Before you begin: Polygon Angle Sums

Understand the idea

A central angle picks the same fraction of a circle’s circumference as its fraction of a full $\displaystyle 360^{\circ}$ turn. The radius scales circumference linearly.

\displaystyle s = \left(\frac{\theta}{360^{\circ}}\right) \cdot 2\, \pi\, r

Choose and carry out a method

Compute angle/360 times $\displaystyle 2\, \pi\, r$ . When the prompt asks for the coefficient of $\displaystyle \pi$ , enter the rational coefficient only.

Check the reasoning

$\displaystyle A\, 180^{\circ}$ arc is half the circumference. Arc length has length units; a sector’s area has square units.

WORKED EXAMPLE 1

A circle has radius 4 and a central angle of $\displaystyle 60^{\circ}$ . The intercepted arc length is $\displaystyle k\, \pi$ . Find k.

Use the central angle as a fraction of a full revolution.
Arc length= $\displaystyle \left(\frac{60}{360}\right) \cdot 2\, \pi \cdot 4$ .
The coefficient of $\displaystyle \pi$ is $\displaystyle \frac{4}{3}$ . This is an arc length, not a sector area.

$\displaystyle \frac{4}{3}$

WORKED EXAMPLE 2

A circle has radius 5 and a central angle of $\displaystyle 90^{\circ}$ . The intercepted arc length is $\displaystyle k\, \pi$ . Find k.

Use the central angle as a fraction of a full revolution.
Arc length= $\displaystyle \left(\frac{90}{360}\right) \cdot 2\, \pi \cdot 5$ .
The coefficient of $\displaystyle \pi$ is $\displaystyle \frac{5}{2}$ . This is an arc length, not a sector area.

$\displaystyle \frac{5}{2}$

Common pitfalls

Possible mix-up: Use $\displaystyle \pi\, r^{2}$ to find arc length.

That measures an area; use circumference for an arc.

Possible mix-up: A correct numerical answer alone explains the method.

State the governing relationship and check the conditions described above.

Explain it to yourself

What changes if the radius doubles but the angle stays fixed?

Preview the eight practice prompts

Problem 1
A circle has radius 7 and a central angle of $\displaystyle 150^{\circ}$ . The intercepted arc length is $\displaystyle k\, \pi$ . Find k.
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Problem 2
A circle has radius 8 and a central angle of $\displaystyle 30^{\circ}$ . The intercepted arc length is $\displaystyle k\, \pi$ . Find k.
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Problem 3
A circle has radius 9 and a central angle of $\displaystyle 60^{\circ}$ . The intercepted arc length is $\displaystyle k\, \pi$ . Find k.
Practise this exact problem →
Problem 4
A circle has radius 10 and a central angle of $\displaystyle 90^{\circ}$ . The intercepted arc length is $\displaystyle k\, \pi$ . Find k.
Practise this exact problem →
Problem 5
A circle has radius 11 and a central angle of $\displaystyle 120^{\circ}$ . The intercepted arc length is $\displaystyle k\, \pi$ . Find k.
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Problem 6
A circle has radius 12 and a central angle of $\displaystyle 150^{\circ}$ . The intercepted arc length is $\displaystyle k\, \pi$ . Find k.
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Problem 7New context
A wheel of radius 13 cm turns through $\displaystyle 30^{\circ}$ . A point on its rim travels $\displaystyle k\, \pi$ cm along the circle. Find k.
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Problem 8New context
A wheel of radius 14 cm turns through $\displaystyle 60^{\circ}$ . A point on its rim travels $\displaystyle k\, \pi$ cm along the circle. Find k.
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Open stage GEO 3.2 in the student workspace →

STAGE GEO 3.3 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Intersecting Chords

Useful preparation: Arcs and Central Angles

Goal: Understand and apply intersecting chords.

Before you begin: Arcs and Central Angles

Understand the idea

Two chords intersecting inside a circle form similar triangles. Corresponding side ratios imply that the two segment lengths on one chord have the same product as those on the other.

\displaystyle a \cdot b = c \cdot d

Choose and carry out a method

Pair the two parts of each whole chord. Set their products equal and divide by the known segment next to the unknown one.

Check the reasoning

Keep segment lengths distinct from full chord lengths. The recovered segment must be positive and satisfy the product equation.

WORKED EXAMPLE 1

Two chords intersect inside a circle. One has segments 3 and 8; the other has segments 4 and x. Find the segment labeled x.

For intersecting chords, the two segment products are equal.
$\displaystyle 3 \cdot 8 = 4 \cdot x$ , so $\displaystyle x = \frac{24}{4}$ .
The missing segment is 6; the two products agree.

WORKED EXAMPLE 2

Two chords intersect inside a circle. One has segments 4 and 10; the other has segments 5 and x. Find the segment labeled x.

For intersecting chords, the two segment products are equal.
$\displaystyle 4 \cdot 10 = 5 \cdot x$ , so $\displaystyle x = \frac{40}{5}$ .
The missing segment is 8; the two products agree.

Common pitfalls

Possible mix-up: Multiply neighboring segments from different chords.

Each product uses the two pieces of one chord.

Possible mix-up: A correct numerical answer alone explains the method.

State the governing relationship and check the conditions described above.

Explain it to yourself

Sketch the similar triangles that explain the product relationship.

Preview the eight practice prompts

Problem 1
Two chords intersect inside a circle. One has segments 6 and 14; the other has segments 7 and x. Find the segment labeled x.
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Problem 2
Two chords intersect inside a circle. One has segments 7 and 16; the other has segments 8 and x. Find the segment labeled x.
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Problem 3
Two chords intersect inside a circle. One has segments 8 and 18; the other has segments 9 and x. Find the segment labeled x.
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Problem 4
Two chords intersect inside a circle. One has segments 9 and 20; the other has segments 10 and x. Find the segment labeled x.
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Problem 5
Two chords intersect inside a circle. One has segments 10 and 22; the other has segments 11 and x. Find the segment labeled x.
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Problem 6
Two chords intersect inside a circle. One has segments 11 and 24; the other has segments 12 and x. Find the segment labeled x.
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Problem 7New context
Two straight supports, each joining two points on a circular frame, cross inside the circle. One support is split into 12 cm and 26 cm; the second has one piece 13 cm long. Find the other piece in centimeters.
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Problem 8New context
Two straight supports, each joining two points on a circular frame, cross inside the circle. One support is split into 13 cm and 28 cm; the second has one piece 14 cm long. Find the other piece in centimeters.
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Open stage GEO 3.3 in the student workspace →

Browse another level →

Polygons and Circles

Polygon Angle Sums

Understand the idea

Choose and carry out a method

Check the reasoning

Find the sum of the interior angles of a convex 6-gon, in degrees.

Find the sum of the interior angles of a convex 7-gon, in degrees.

Common pitfalls

Explain it to yourself

Arcs and Central Angles

Understand the idea

Choose and carry out a method

Check the reasoning

A circle has radius 4 and a central angle of 60∘\displaystyle 60^{\circ}60∘. The intercepted arc length is k π\displaystyle k\, \pikπ. Find k.

A circle has radius 5 and a central angle of 90∘\displaystyle 90^{\circ}90∘. The intercepted arc length is k π\displaystyle k\, \pikπ. Find k.

Common pitfalls

Explain it to yourself

Intersecting Chords

Understand the idea

Choose and carry out a method

Check the reasoning

Two chords intersect inside a circle. One has segments 3 and 8; the other has segments 4 and x. Find the segment labeled x.

Two chords intersect inside a circle. One has segments 4 and 10; the other has segments 5 and x. Find the segment labeled x.

Common pitfalls

Explain it to yourself

A circle has radius 4 and a central angle of $\displaystyle 60^{\circ}$ . The intercepted arc length is $\displaystyle k\, \pi$ . Find k.

A circle has radius 5 and a central angle of $\displaystyle 90^{\circ}$ . The intercepted arc length is $\displaystyle k\, \pi$ . Find k.