The Types of Quadrilaterals and How They Relate

Four-sided shapes are sorted into families by shared traits like parallel sides, equal sides, and right angles. Every member of a family has all of that family's properties, so once you know a shape is a parallelogram, every parallelogram rule applies to it. We start with a frame of four rods bolted loosely at the corners.

Problem

A picture frame is built from two

8

inch rods and two

5

inch rods, hinged loosely at the corners. Push on one corner and the rectangle leans into the slanted shape above, then leans further with another push. The angles keep changing. What is the perimeter of the leaning shape, in inches?

Show a hint

Leaning the frame does not change the rods, only the angles at the hinges.
Add up the four rod lengths, $8 + 5 + 8 + 5$ .

Show the full solution

Add up the four rod lengths.

8 + 5 + 8 + 5 = 26

inches. Leaning the frame changes the angles at the hinges, never the rods, so every position has this same perimeter.

Problem

In one leaning position, the hinge at the bottom left of the frame reads

70

degrees. The bottom rod and the top rod stay parallel, and so do the two side rods. Using what you know about parallel lines, how many degrees is the angle at the bottom right hinge?

Show a hint

The two side rods are parallel, and the bottom rod crosses both of them, so the two bottom angles are co-interior angles.
Co-interior angles between parallel lines add to $180$ degrees.

Show the full solution

The two bottom hinge angles add to

180

degrees, so the right one is

180 - 70 = 110

degrees. The bottom rod crosses both parallel side rods, which is what makes those two angles co-interior.

Problem

A parallelogram-shaped stencil has a base of

10

cm, slanted sides of

7

cm, and a height of

6

cm measured straight up from the base. Snip the overhanging triangle off one end and slide it across to plug the gap at the other end. The pieces form a rectangle. What is the stencil's area in square centimeters?

Show a hint

The rectangle you build has the same base and the same height as the stencil.
A $10$ by $6$ rectangle has area $10 \times 6$ .

Show the full solution

The rectangle is

10

6

, so the area is

10 \times 6 = 60

square centimeters. The snipped triangle fills the gap exactly, so nothing is gained or lost, and the

7

cm slant never enters the calculation.

Problem

A solar panel shaped like a parallelogram has sides of

12

m and

5

m. The perpendicular distance between its two

12

m edges is

4

m. An installer multiplies

12 \times 5 = 60

and quotes the area as

60

square meters. What is the actual area, in square meters?

Show a hint

Area needs the base times the height, and the height is the perpendicular distance, not the slanted side.
The base is $12$ and the height is $4$ .

Show the full solution

Area is base times height,

12 \times 4 = 48

square meters. The

5

m side is slanted, so it is longer than the true height of

4

m, and using it overquotes by

12

square meters.

Problem

A club logo is drawn by marking a horizontal segment of length

12

and a vertical segment of length

9

that cross at each other's midpoints, then connecting the four endpoints. The result is the rhombus above, a quadrilateral with four equal sides. It sits inside a dashed

12

9

rectangle, and in each quarter of that rectangle the logo covers exactly half. What is the area of the logo?

Show a hint

The dashed rectangle has area $12 \times 9 = 108$ .
Each quarter-rectangle is cut along its diagonal, so the logo takes exactly half of the whole rectangle.

Show the full solution

The logo covers half the dashed rectangle, and

12 \times 9 = 108

, so the area is

54

. Each quarter of the rectangle is cut along its diagonal, which is why the logo takes exactly half.

Problem

A rhombus-shaped garden plot has an area of

40

square feet. One diagonal measures

10

feet. Using the pattern you just discovered, how many feet long is the other diagonal?

Show a hint

A rhombus fills half its bounding rectangle, so area $=$ (diagonal $\times$ diagonal) $\div 2$ .
Solve $10 \times d \div 2 = 40$ .

Show the full solution

The diagonals multiply to twice the area,

10 \times d = 80

, so

d = 8

feet.

Problem

A highway warning sign is a rhombus with diagonals of

12

inches and

16

inches. The diagonals cross at right angles at their midpoints, cutting the sign into four identical right triangles. Use one of those triangles and the Pythagorean Theorem to find the side length of the sign, in inches.

Show a hint

Each right triangle has legs equal to half of each diagonal, $6$ and $8$ .
Find the hypotenuse of a $6$ - $8$ right triangle.

Show the full solution

Half-diagonals of

6

and

8

form the legs, and the sign's side is the hypotenuse. Since

6^{2} + 8^{2} = 36 + 64 = 100

, the side is

\sqrt{100} = 10

inches.

Problem

A square napkin is folded once along its diagonal, and the crease measures

10

inches. A square is a rhombus whose diagonals are equal, so both diagonals of the napkin are

10

. What is the napkin's area in square inches?

Show a hint

Use the rhombus area tool with $d_{1} = d_{2} = 10$ .
Half of $10 \times 10$ .

Show the full solution

With equal diagonals of

10

, the rhombus tool gives

\frac{1}{2} \times 10 \times 10 = 50

square inches. You met this napkin in the last lesson, where the side came out to

\frac{10}{\sqrt{2}}

, and squaring that side gives the same

50

Problem

A kitchen backsplash uses trapezoid tiles with parallel edges of

5

inches and

11

inches, set

4

inches apart. Tiles snap together in pairs, one flipped upside down as shown, forming a parallelogram. What is the area of that two-tile parallelogram, in square inches?

Show a hint

The flipped tile lays its $5$ inch edge alongside the other tile's $11$ inch edge, so the parallelogram's base is $5 + 11$ .
Base $16$ , height $4$ , area $= 16 \times 4$ .

Show the full solution

Snapped together, the pair forms a parallelogram with base

5 + 11 = 16

and the same

4

inch height, so the area is

16 \times 4 = 64

square inches.

Problem

One single tile is half of that two-tile parallelogram. Check the shortcut on a bigger tile. A trapezoid window has parallel edges of

9

feet and

13

feet, set

6

feet apart. What is its area in square feet?

Show a hint

Two copies would form a parallelogram of base $9 + 13 = 22$ and height $6$ , and the window is half of it.
Compute $\frac{1}{2} \times 22 \times 6$ .

Show the full solution

Doubling the window gives a parallelogram of area

22 \times 6 = 132

, and one window is half of that,

\frac{1}{2} \times 132 = 66

square feet. In one line, area

= \frac{1}{2} (9 + 13) (6)

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Practice

Here are four claims. (a) Every square is a rhombus. (b) Every rhombus is a square. (c) Every rectangle is a parallelogram. (d) A quadrilateral can have exactly three right angles. How many of the four claims are true?

Show the solution

(a) is true, a square's four equal sides make it a rhombus. (b) is false, a leaning rhombus has no right angles. (c) is true, right angles do not remove parallel sides. (d) is false, three right angles use up

270

degrees and the fourth must be

90

, giving four right angles, never exactly three. That makes

2

true claims.

Practice

One angle of a parallelogram measures

48

degrees. How many degrees is the largest angle of the parallelogram?

Show the solution

A neighbor of the

48

degree angle measures

180 - 48 = 132

degrees, and opposite angles repeat the pair. The largest is

132

Practice

A rhombus has a perimeter of

52

cm, and one of its diagonals measures

24

cm. Use the four right triangles inside the rhombus to find the length of the other diagonal, in centimeters.

Show the solution

Each side is

52 \div 4 = 13

, and the half-diagonals are the legs of a right triangle with hypotenuse

13

. One leg is

24 \div 2 = 12

, so the other satisfies

a^{2} + 1 2^{2} = 1 3^{2}

, giving

a^{2} = 169 - 144 = 25

and

a = 5

. The full diagonal is

2 \times 5 = 10

cm.

Practice

A stained-glass pane is a rhombus with diagonals of

14

inches and

6

inches. What is its area in square inches?

Show the solution

Half the product of the diagonals is

\frac{1}{2} \times 14 \times 6 = 42

square inches.

Practice

A square coaster has a diagonal of

6

inches. What is its area in square inches?

Show the solution

Treat the square as a rhombus with equal diagonals. Its area is

\frac{1}{2} \times 6 \times 6 = 18

square inches, no side length needed.

Practice

A parallelogram-shaped field has an area of

91

square meters and a base of

13

meters. What is its height, in meters?

Show the solution

Height

= 91 \div 13 = 7

meters.

Practice

The cross-section of a drainage channel is a trapezoid with parallel edges of

4

feet and

10

feet, set

5

feet apart. What is the area of the cross-section in square feet?

Show the solution

The average base is

\frac{4 + 10}{2} = 7

, and

7 \times 5 = 35

square feet.

Practice

A trapezoid banner has an area of

60

square inches, a height of

6

inches, and one base of

7

inches. How many inches long is the other base?

Show the solution

The sum of the bases is

\frac{2 \times 60}{6} = 20

, so the other base is

20 - 7 = 13

inches.

Practice

A pixel-art gem on a screen has corners at

(0, 4)

(6, 0)

(12, 4)

, and

(6, 8)

. Its diagonals are the horizontal segment from

(0, 4)

(12, 4)

and the vertical segment from

(6, 0)

(6, 8)

. What is the gem's area in square pixels?

Show the solution

The diagonals measure

12

and

8

, cross at

(6, 4)

, and are perpendicular, so the area is

\frac{1}{2} \times 12 \times 8 = 48

square pixels.

Practice

A tile has four

6

cm sides and at least one right angle. Explain to yourself which family the tile must belong to, then find its area in square centimeters.

Show the solution

Four equal sides make the tile a rhombus, hence a parallelogram. One

90

degree angle forces each neighbor to

180 - 90 = 90

, so all four angles are right and the tile is a square. Its area is

6^{2} = 36

square centimeters.

Families of Quadrilaterals

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