28. Area and Length of Complex Shapes

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Hexagon

In this section you will be able to find the angles and the area of a regular hexagon by cutting it into equilateral triangles.

Bees build their honeycomb out of hexagons. Look closely at one cell: you can cut it into 6 triangles that are all exactly the same.

A regular hexagon has 6 equal sides and 6 equal angles. The center is the same distance from every vertex. So the 6 triangles from the center are all equilateral, with side s.

Look:once you know one equilateral triangle, you know the whole hexagon: area, angles and diagonals.

Exploreplay first, formulas later

Try this: Tap the pieces of the 6-triangle split. Is every piece the same size? Each one is an equilateral triangle with side s.
Try this: Press 2 trapezoids, then Rectangle + 2. Does the area change? Turn on Show the formula and compare the three build-ups.
Try this: Set s to 1, then 2, then 3. The area goes up 4 times, then 9 times. Why do squares show up?
Try this: Press All diagonals. How many pieces? Which piece is the smallest?

Name it

Theorem 28.1.1 · Regular Hexagon

Sum of interior angle of a regular hexagon $= (6-2)\cdot 180 = 720$

Interior angle of a regular hexagon $= \dfrac{(6-2)}{6}\cdot 180 = 120$

Exterior angle of a regular hexagon $= \dfrac{360}{6} = 60$

Area of a regular hexagon $= 6\cdot\dfrac{\sqrt3}{4}s^2$

Length of the diagonal of a regular hexagon $= 2s$

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Example 28.1

EMCC

Vincent the Bug draws all the diagonals of a regular hexagon with area 720, splitting it into many pieces. Compute the area of the smallest piece.

1
A hexagon has 9 diagonals. After you draw them all, how many pieces are there? Count them in the picture.

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Example 28.2

EMCC

In equilateral triangle $ABC$ , points $P$ and $R$ lie on segment $AB$ , points $I$ and $M$ lie on segment $BC$ , and points $E$ and $S$ lie on segment $CA$ such that $PRIMES$ is a equiangular hexagon. Given that $AB = 11$ , $PS = 2$ , $RI = 3$ , and $ME = 5$ , compute the area of hexagon $PRIMES$ .

1
Equiangular means every angle of the hexagon is the same. What is each angle?

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Problem 28.5.4

AMC 8

A unit hexagram is composed of a regular hexagon of side length 1 and its 6 equilateral triangular extensions, as shown in the diagram. What is the ratio of the area of the extensions to the area of the original hexagon?

Problem 28.5.13

AMC 8

An equilateral triangle and a regular hexagon have equal perimeters. If the area of the triangle is 4, what is the area of the hexagon?

Contest cornerreal contest problems

Problem 28.5.9

AMC 8

In the given figure hexagon $ABCDEF$ is equiangular, $ABJI$ and $FEHG$ are squares with areas 18 and 32 respectively, $\triangle JBK$ is equilateral and $FE = BC$ . What is the area of $\triangle KBC$ ?

Problem 28.5.16(additional)

AMC 10

A regular hexagon of side length 1 is inscribed in a circle. Each minor arc of the circle determined by a side of the hexagon is reflected over that side. What is the area of the region bounded by these 6 reflected arcs?

Checkpoint for section 28.1

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

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Octagon

In this section you will be able to make a regular octagon by cutting the corners off a square, and find its angles and area.

A stop sign is a regular octagon. You can make one from a square piece of card: just snip off the four corners. But how much do you snip?

Each corner you cut is an isosceles right triangle, a 45-45-90 triangle. If its legs are x, the cut edge is x√2. The octagon is regular when every cut edge matches what is left of the square’s sides.

Look:the octagon’s area is the square minus four small triangles. That is where 2(1 + √2)s² comes from.

Exploreplay first, formulas later

Try this: Drag the slider. Watch the slanted sides and the straight sides. When are they equal?
Try this: Press Make it regular. Each slanted side is x√2. Why does x√2 + 2x = 10 have to be true?
Try this: The area is the square minus 4 corner triangles. Check the readout: is it 100 − 2x²?

Name it

Theorem 28.2.1 · Regular Octagon

Sum of interior angle of a regular octagon $= (8-2)\cdot 180 = 1080$

Interior angle of a regular octagon $= \dfrac{(8-2)}{8}\cdot 180 = 135$

Exterior angle of a regular octagon $= \dfrac{360}{8} = 45$

Area of a regular octagon $= 2(1+\sqrt2)s^2$

Contest cornerreal contest problems

Problem 28.5.1

AMC 8

Point $O$ is the center of the regular octagon $ABCDEFGH$ , and $X$ is the midpoint of the side $\overline{AB}$ . What fraction of the area of the octagon is shaded?

Checkpoint for section 28.2

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

28.30%

Area of Complex Shapes

In this section you will be able to find the area of strange shapes by adding nice pieces and subtracting the extra ones.

How much paint covers an arched doorway wall? It isn’t a rectangle or a circle. It’s a rectangle plus half a circle, minus the door!

You already know the areas of rectangles, triangles and circles. A complex shape is just these pieces glued together, or with pieces cut away. Add the pieces you need and subtract the ones that don’t belong.

Look:sometimes it is easier to start with a bigger, nicer shape and take away what is extra.

Exploreplay first, formulas later

Try this: Tap the half circle to add it. Then tap the door twice so it is subtracted. What is the area of the arch?
Try this: Look at the running total. Which parts have π in them, and which don't?
Try this: Press Clear and build the area in a different order. Do you get the same answer?

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Concept 28.3.1 · Tricks to finding the area of complex shapes

Divide the shape into “nicer” areas which are easier to calculate
Extend Lines
- You generally want to extend lines when they form nicer shapes/areas to work with, such as triangles
Break up areas
- A common way to do so is to drop altitudes as doing so generally allows you to form right triangles

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Example 28.3

EMCC

In rectangle $TRIG$ , points $A$ and $L$ lie on sides $TG$ and $TR$ respectively such that $TA = AG$ and $TL = 2LR$ . Diagonal $GR$ intersects segments $IL$ and $IA$ at $B$ and $E$ respectively. Suppose that the area of the convex pentagon with vertices $TABLE$ is equal to 21. What is the area of $TRIG$ ?

Drawn with TRIG = 8 × 7. Tap the pieces to build the pentagon.

1
You don't know the rectangle's size, so find every area as a fraction of $TRIG$ . What fraction of $TRIG$ is triangle $TRG$ ?

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Problem 28.5.2

AMC 8

In the figure, the outer equilateral triangle has area 16, the inner equilateral triangle has area 1, and the three trapezoids are congruent. What is the area of one of the trapezoids?

Problem 28.5.6

AMC 8

A decorative window is made up of a rectangle with semicircles on either end. The ratio of $AD$ to $AB$ is $3:2$ , and $AB$ is 30 inches. What is the ratio of the area of the rectangle to the combined areas of the semicircles?

Contest cornerreal contest problems

Problem 28.5.10

AMC 8

In the figure, $ABCD$ is a rectangle and $EFGH$ is a parallelogram. Using the measurements given in the figure, what is the length $d$ of the segment that is perpendicular to $\overline{HE}$ and $\overline{FG}$ ?

Problem 28.5.11

AMC 8

Two $4\times 4$ squares intersect at right angles, bisecting their intersecting sides, as shown. The circle's diameter is the segment between the two points of intersection. What is the area of the shaded region created by removing the circle from the squares?

Problem 28.5.12

AMC 8

A square with side length 2 and a circle share the same center. The total area of the regions that are inside the circle and outside the square is equal to the total area of the regions that are outside the circle and inside the square. What is the radius of the circle?

Problem 28.5.15(additional)

MATHCOUNTS

In the figure, a kite with two right angles is circumscribed by a circle. The circle is then circumscribed by a square such that the diagonals of the kite are parallel to the sides of the square as shown. The longer sides of the kite each have length $x$ meters. The sides of the kite divide the square into three regions, shaded dark gray, white and light gray as shown, with each region of area 2. If $x^2 = a + \sqrt{b}$ , what is the value of $a + b$ ?

Checkpoint for section 28.3

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

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Length of Complex Shapes

In this section you will be able to find perimeters and other lengths by splitting them into segments and arcs you can measure.

An ant walks once around a staircase-shaped garden. How far does it walk? You can add up every little edge, or you can find a shortcut!

A length around a complex shape is made of straight segments and arcs. Find each piece, then add. Right angles let you use the Pythagorean theorem. Equal angles often mean equal lengths.

Look:dropping an altitude turns a slanted side into the long side of a right triangle.

Exploreplay first, formulas later

Try this: Tap each edge of the staircase to add it. What is the total? Compare it with the dashed square's perimeter.
Try this: Each step's top edge slides up to the top of the square. Each riser slides out to the right. Why does that keep the length?
Try this: Press Show me on the card below. Which lengths do the altitudes help you find?

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Concept 28.4.1 · Finding Length of Complex Shapes

Having equal angles means equal lengths and vice versa
Be on the lookout for 90 degree angles, as you can use Pythagorean theorem
Split the length into multiple components by using some of these techniques
- Drawing extra lines
- Dropping Altitudes
- Extending lines to create similar triangles, special triangles, etc. and then subtracting the extra length

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Example 28.4

EMCC

Two circles, with radius 6 and radius 8, are externally tangent to each other. Two more circles, of radius 7, are placed on either side of this configuration, so that they are both externally tangent to both of the original two circles. Out of these 4 circles, what is the maximum distance between any two centers?

1
When two circles touch from the outside, the distance between centers is the sum of the radii. How far apart are the centers of the radius-6 and radius-8 circles?

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Example 28.5

EMCC

David decides he wants to join the West African Drumming Ensemble, and thus he goes to the store and buys three large cylindrical drums. In order to ensure none of the drums drop on the way home, he ties a rope around all of the drums at their mid sections so that each drum is next to the other two. Suppose that each drum has a diameter of 3.5 feet. David needs $m$ feet of rope. Given that $m = a\pi + b$ , where $a$ and $b$ are rational numbers, find $a + b$ .

1
The rope is made of straight parts and curved parts. What is the radius of each drum?

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Problem 28.5.3

AMC 8

The letter T is formed by placing two $2\times 4$ inch rectangles next to each other, as shown. What is the perimeter of the T, in inches?

Problem 28.5.5

AMC 8

The area of polygon $ABCDEF$ is 52 with $AB = 8$ , $BC = 9$ and $FA = 5$ . What is $DE + EF$ ?

Problem 28.5.8

AMC 8

What is the perimeter of trapezoid $ABCD$ ?

Contest cornerreal contest problems

Problem 28.5.7

AMC 8

Triangle $ABC$ is an isosceles triangle with $\overline{AB} = \overline{BC}$ . Point $D$ is the midpoint of both $\overline{BC}$ and $\overline{AE}$ , and $\overline{CE}$ is 11 units long. Triangle $ABD$ is congruent to triangle $ECD$ . What is the length of $\overline{BD}$ ?

Problem 28.5.14

AMC 8

Angle $ABC$ of $\triangle ABC$ is a right angle. The sides of $\triangle ABC$ are the diameters of semicircles as shown. The area of the semicircle on $\overline{AB}$ equals $8\pi$ , and the arc of the semicircle on $\overline{AC}$ has length $8.5\pi$ . What is the radius of the semicircle on $\overline{BC}$ ?

Checkpoint for section 28.4

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

28.5

Practice Problems

In this section you will be able to mix every trick in the chapter on a tough contest problem.

Three paper squares, turned and stacked, make a 24-sided star. Can you find its area? Look for the symmetry first!

Want every problem from this chapter on one page? Open the problem set.

Contest cornerreal contest problems

Problem 28.5.17(additional)

AMC 10

Three identical square sheets of paper each with side length 6 are stacked on top of each other. The middle sheet is rotated clockwise $30^\circ$ about its center and the top sheet is rotated clockwise $60^\circ$ about its center, resulting in the 24-sided polygon shown in the figure below. The area of this polygon can be expressed in the form $a - b\sqrt{c}$ , where $a$ , $b$ , and $c$ are positive integers, and $c$ is not divisible by the square of any prime. What is $a + b + c$ ?