Geometry · Chapter 35

Polygons

Regular hexagons, octagons and pentagons hide simpler shapes inside: six equilateral triangles, a square with its corners cut off, and the golden ratio.

What's this chapter about?

35.0 Polygon Angles
35.1 Hexagon
35.2 Octagon
35.3 Pentagons
35.4 Practice Problems

You'll need

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35.00%

Polygon Angles

In this section you will be able to find the angle sum of any polygon and each angle of a regular one, using the 360° turn around its edge.

Walk once around any polygon and you end up facing the way you started. You turned exactly once: $360^\circ$ , no matter how many corners.

That one fact gives every angle rule in this chapter. A regular $n$ -gon turns the same amount at each corner, so each exterior angle is $\frac{360^\circ}{n}$ and each interior angle is $180^\circ$ minus that.

You met these rules in Chapter 29. Here they are the starting point: the hexagon, octagon and pentagon each have their own tricks built on them.

Look:When you know the interior angle, find the exterior angle first:

180^\circ - 156^\circ = 24^\circ

, so

\frac{360}{24} = 15

sides.

Exploreplay first, formulas later

Try this: Walk the edge of a pentagon, then an octagon, then a 12-gon. How much do you turn in total each time?
Try this: At one corner, add the interior and the exterior angle. Why is the answer always 180°?
Try this: Turn on the triangles from one corner. How many triangles does an n-gon split into?

Name it

Theorem 35.0.1

For a polygon with $n$ sides:

\text{Sum of interior angles} = (n-2)\cdot 180^\circ

\text{Interior angle of a regular polygon} = \frac{(n-2)}{n}\cdot 180^\circ

\text{Exterior angle of a regular polygon} = \frac{360^\circ}{n}

Fact 35.0.2 · Important Interior Angles

Interior angles of the regular polygons you meet most:

Number of sides	Interior angle
3	$60^\circ$
4	$90^\circ$
5	$108^\circ$
6	$120^\circ$
8	$135^\circ$
9	$140^\circ$
10	$144^\circ$

Checkpoint for section 35.0

Get 2 of 3 right to master this section. One try each, no hints.

35.10%

Hexagon

In this section you will be able to cut a regular hexagon into six equilateral triangles to get its area and diagonals, and handle hexagons with only equal angles.

A regular hexagon is six equilateral triangles in a ring. Spot that ring and almost every hexagon question turns into a question about equilateral triangles.

Join the center to the six corners. Each triangle has a $60^\circ$ angle at the center and two equal sides, so it is equilateral. Its side is the hexagon’s side $s$ , and the long diagonal is two of those sides in a line.

An equiangular hexagon has all angles $120^\circ$ but sides that can differ. Extend every other side and the corners fill in with equilateral triangles: the hexagon becomes a big equilateral triangle with three corners cut off.

Look:In an equiangular hexagon, opposite sides are parallel, and the last two sides are forced by the first four.

Exploreplay first, formulas later

Try this: Pick “6 triangles”. Each piece is equilateral with side s. Why does that give an area of 6 · √3/4 · s²?
Try this: Turn on Long diagonal. Which two triangle sides does it run along? So why is it 2s?
Try this: Now open the figure of Problem 35.4.2. Extend every other side of an equiangular hexagon: what shape appears?

Name it

Theorem 35.1.1 · Regular Hexagon

For a regular hexagon with side $s$ :

\text{Sum of interior angles} = (6-2)\cdot 180^\circ = 720^\circ

\text{Interior angle} = \frac{(6-2)}{6}\cdot 180^\circ = 120^\circ

\text{Exterior angle} = \frac{360^\circ}{6} = 60^\circ

\text{Area} = 6\cdot\frac{\sqrt{3}}{4}s^2

\text{Long diagonal} = 2s

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Example 35.1

AMC 10

Six regular hexagons surround a regular hexagon of side length $1$ as shown. What is the area of $\triangle ABC$ ?

1
Let $O$ be the center of the middle hexagon. Walk from $O$ straight to $A$ . How long is $OA$ ?

Predict first:
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Example 35.2

AIME

In convex hexagon $ABCDEF$ , all six sides are congruent, $\angle A$ and $\angle D$ are right angles, and $\angle B, \angle C, \angle E,$ and $\angle F$ are congruent. The area of the hexagonal region is $2116(\sqrt{2}+1)$ . Find $AB$ .

1
The six angles add to $720^\circ$ . Two of them are $90^\circ$ and the other four are equal. How big is each of those four?

Predict first:
Your prediction

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Example 35.3

AIME

A hexagon that is inscribed in a circle has side lengths $22, 22, 20, 22, 22,$ and $20$ in that order. The radius of the circle can be written as $p + \sqrt{q}$ , where $p$ and $q$ are positive integers. Find $p + q$ .

1
Join the center $O$ to every vertex. Call the central angle over each side $22$ by $2\alpha$ and over each side $20$ by $2\beta$ . The six central angles fill the circle. What is $2\alpha + \beta$ , in degrees?

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Problem 35.4.9(additional)

AMC 10

Equilateral $\triangle ABC$ has side length $1$ , and squares $ABDE$ , $BCHI$ , $CAFG$ lie outside the triangle. What is the area of hexagon $DEFGHI$ ?

Contest cornerreal contest problems

Problem 35.4.1

AMC 10

A wire is cut into two pieces, one of length $a$ and the other of length $b$ . The piece of length $a$ is bent to form an equilateral triangle, and the piece of length $b$ is bent to form a regular hexagon. The triangle and the hexagon have equal area. What is $\frac{a}{b}$ ?

Problem 35.4.8(additional)

AMC 10

Spot's doghouse has a regular hexagonal base that measures one yard on each side. He is tethered to a vertex with a two-yard rope. What is the area, in square yards, of the region outside of the doghouse that Spot can reach?

Problem 35.4.5

AMC 10

In regular hexagon $ABCDEF$ , points $W$ , $X$ , $Y$ , and $Z$ are chosen on sides $\overline{BC}$ , $\overline{CD}$ , $\overline{EF}$ , and $\overline{FA}$ respectively, so lines $AB$ , $ZW$ , $YX$ , and $ED$ are parallel and equally spaced. What is the ratio of the area of hexagon $WCXYFZ$ to the area of hexagon $ABCDEF$ ?

Problem 35.4.2

AMC 10

This problem also appears in the book as 29.2.1.

Let $ABCDEF$ be an equiangular hexagon. The lines $AB, CD,$ and $EF$ determine a triangle with area $192\sqrt{3}$ , and the lines $BC, DE,$ and $FA$ determine a triangle with area $324\sqrt{3}$ . The perimeter of hexagon $ABCDEF$ can be expressed as $m + n\sqrt{p}$ , where $m, n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime. What is $m + n + p$ ?

Problem 35.4.11(additional)

AMC 10/12

Equiangular hexagon $ABCDEF$ has side lengths $AB = CD = EF = 1$ and $BC = DE = FA = r$ . The area of $\triangle ACE$ is $70\%$ of the area of the hexagon. What is the sum of all possible values of $r$ ?

Problem 35.4.13(additional)

AIME

Let $ABCDEF$ be an equiangular hexagon such that $AB = 6$ , $BC = 8$ , $CD = 10$ , and $DE = 12$ . Denote by $d$ the diameter of the largest circle that fits inside the hexagon. Find $d^2$ .

Checkpoint for section 35.1

Get 2 of 3 right to master this section. One try each, no hints.

35.20%

Octagon

In this section you will be able to see a regular octagon as a square with four corners cut off, and use that to find its sides and area.

Cut four equal corners off a square. At exactly one size of cut, all eight sides match and you get a regular octagon.

Each corner you cut is an isosceles right triangle. If the octagon’s side is $s$ , the corner’s legs are $\frac{s}{\sqrt2}$ , so the square’s side is $s + s\sqrt2 = s(1+\sqrt2)$ .

Now the octagon’s area is the square minus four small triangles: $2(1+\sqrt2)s^2$ . Rectangles and squares drawn on its corners are easy to measure in the same picture.

Look:A regular octagon’s interior angle is

135^\circ = 90^\circ + 45^\circ

: that

45^\circ

is why the corners are 45-45-90 triangles.

Exploreplay first, formulas later

Try this: Drag the corner cut until all eight sides match. What is the corner leg compared with the octagon's side s?
Try this: Press “Make it regular”. Check that the area matches 2(1 + √2)s².
Try this: For the regular octagon, the area is the square minus four corners. Which piece would you shade to get exactly half?

Name it

Theorem 35.2.1 · Regular Octagon

For a regular octagon with side $s$ :

\text{Sum of interior angles} = (8-2)\cdot 180^\circ = 1080^\circ

\text{Interior angle} = \frac{(8-2)}{8}\cdot 180^\circ = 135^\circ

\text{Exterior angle} = \frac{360^\circ}{8} = 45^\circ

\text{Area} = 2(1+\sqrt{2})s^2

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Example 35.4

AMC 10

A regular octagon $ABCDEFGH$ has an area of one square unit. What is the area of the rectangle $ABEF$ ?

1
See the octagon as a square with its four corners cut off. Each corner is an isosceles right triangle whose hypotenuse is a side $s$ of the octagon. How long is each leg?

Predict first:
Your prediction

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Example 35.6

AMC 12

Regular octagon $ABCDEFGH$ has area $n$ . Let $m$ be the area of quadrilateral $ACEG$ . What is $\dfrac{m}{n}$ ?

1
Only the ratio matters, so take side $1$ . What is $n$ , the octagon's area?

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Problem 35.4.7(additional)

AMC 10

A regular octagon is formed by cutting an isosceles right triangle from each of the corners of a square with sides of length $2000$ . What is the length of each side of the octagon?

Checkpoint for section 35.2

Get 2 of 3 right to master this section. One try each, no hints.

35.30%

Pentagons

In this section you will be able to use the 36-72-72 triangles inside a regular pentagon to find lengths in terms of the golden ratio φ.

Draw all five diagonals of a regular pentagon and a star appears. Every triangle in it is one of just two shapes, and they hide the golden ratio.

Each angle of a regular pentagon is $108^\circ$ . The two diagonals from a corner split it into three $36^\circ$ angles, so the triangles have angles $36^\circ, 72^\circ, 72^\circ$ or $36^\circ, 36^\circ, 108^\circ$ .

Similar 36-72-72 triangles give the key ratio: a diagonal is $\varphi = \frac{1+\sqrt5}{2} \approx 1.618$ times a side. That number satisfies $\varphi^2 = \varphi + 1$ .

Exploreplay first, formulas later

Try this: Turn on each triangle in turn. How many different triangle shapes do the diagonals make?
Try this: Measure a side and a diagonal. What is diagonal ÷ side? Does it change if the side is 1 or 2?
Try this: Measure a star tip and an inner side. How do they compare with φ = (1 + √5)/2?

Watch itpredict, then reveal

Example 35.5

AMC 10

In the figure shown below, $ABCDE$ is a regular pentagon and $AG = 1$ . What is $FG + JH + CD$ ?

1
Each angle of a regular pentagon is $108^\circ$ , and the two diagonals from $A$ cut it into three $36^\circ$ angles. What are the angles of triangle $AFG$ ?

Predict first:
Click a choice or type its letter (A–E).

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Contest cornerreal contest problems

Problem 35.4.4

AMC 10/12

The figure below is constructed from $11$ line segments, each of which has length $2$ . The area of pentagon $ABCDE$ can be written as $\sqrt{m} + \sqrt{n}$ , where $m$ and $n$ are positive integers. What is $m + n$ ?

Problem 35.4.10(additional)

AMC 12

Andrea inscribed a circle inside a regular pentagon, circumscribed a circle around the pentagon, and calculated the area of the region between the two circles. Bethany did the same with a regular heptagon (7 sides). The areas of the two regions were $A$ and $B$ , respectively. Each polygon had a side length of $2$ . Which of the following is true?

Checkpoint for section 35.3

Get 2 of 3 right to master this section. One try each, no hints.

35.4

Practice Problems

In this section you will be able to mix the hexagon, octagon and pentagon tricks on harder problems.

Mixed polygon problems: tangent circles beside a hexagon, a ring of hexagonal blocks, and a hexagon squeezed into a triangle.

Before you compute, find the hidden equilateral triangles, the right isosceles corners, or the 36-72-72 shapes.

Contest cornerreal contest problems

Problem 35.4.3

AMC 10

Two circles lie outside regular hexagon $ABCDEF$ . The first is tangent to $\overline{AB}$ , and the second is tangent to $\overline{DE}$ . Both are tangent to lines $BC$ and $FA$ . What is the ratio of the area of the second circle to that of the first circle?

Problem 35.4.6

AMC 10

Six regular hexagonal blocks of side length $1$ unit are arranged inside a regular hexagonal frame. Each block lies along an inside edge of the frame and is aligned with two other blocks, as shown in the figure below. The distance from any corner of the frame to the nearest vertex of a block is $\frac{3}{7}$ unit. What is the area of the region inside the frame not occupied by the blocks?

Problem 35.4.12(additional)

AIME

Let $\triangle PQR$ be a triangle with $\angle P = 75^\circ$ and $\angle Q = 60^\circ$ . A regular hexagon $ABCDEF$ with side length $1$ is drawn inside $\triangle PQR$ so that side $\overline{AB}$ lies on $\overline{PQ}$ , side $\overline{CD}$ lies on $\overline{QR}$ , and one of the remaining vertices lies on $\overline{RP}$ . There are positive integers $a, b, c,$ and $d$ such that the area of $\triangle PQR$ can be expressed in the form $\frac{a+b\sqrt{c}}{d}$ , where $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a+b+c+d$ .