25. Quadrilaterals · Mastering AMC 8

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Square

In this section you will be able to find the area and perimeter of a square from its side.

A square tile is 5 inches on each side. How many 1-inch squares fit on it? Five rows of five: 25. That is the whole idea of area!

A quadrilateral is a shape with 4 straight sides. The square is the most special one: all 4 sides are equal and all 4 angles are right angles.

Look:if you know just one side of a square, you know everything: its area, its perimeter, even its diagonal.

Exploreplay first, formulas later

Square

All 4 sides equal: yes
4 right angles: yes
Both pairs of opposite sides parallel: yes
At least one pair of opposite sides parallel: yes
Diagonals are equal: yes
Diagonals are perpendicular: yes
Diagonals cut each other in half: yes

Square: area 25, perimeter 20

Try this: Read the side lengths on the square. Count the grid squares inside it. Does the count match side × side?
Try this: Press Rectangle, then Square again. Which boxes in the checklist only a square ticks?
Try this: Turn on Diagonals. In a square, the diagonals are equal AND perpendicular. Which checklist lines say so?

Name it

Theorem 25.1.1 · Area of a Square

Any square with side length $s$ has an area of

s^2

and a perimeter of

4s

Contest cornerreal contest problems

Problem 25.7.2

AMC 8

Points $A, B, C$ and $D$ are midpoints of the sides of the larger square. If the larger square has area 60, what is the area of the smaller square?

Checkpoint for section 25.1

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

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Rectangle

In this section you will be able to find the area and perimeter of a rectangle from its base and height.

A garden is 8 steps long and 4 steps wide. You plant one flower per square step. How many flowers? 8 in each row, 4 rows: 32 flowers.

A rectangle has 4 right angles. Its opposite sides are equal, but the long and short sides can differ. A square is a rectangle whose sides happen to be equal.

Look:in contest problems, rectangles are often built from smaller rectangles. Find one unknown length first, and the rest follows.

Exploreplay first, formulas later

Rectangle

All 4 sides equal: no
4 right angles: yes
Both pairs of opposite sides parallel: yes
At least one pair of opposite sides parallel: yes
Diagonals are equal: yes
Diagonals are perpendicular: no
Diagonals cut each other in half: yes

Rectangle: area 32, perimeter 24

Try this: Drag corner C one step right. The checklist loses its right angles. What shape do you get now?
Try this: Press Rectangle again. Count the grid squares in one row and the number of rows. Does base × height match the area?
Try this: Turn on Show the formula. Which two lengths does the perimeter 2b + 2h use?

Name it

Theorem 25.2.1 · Area of a Rectangle

Any rectangle with base $b$ and height $h$ has an area of

bh

and a perimeter of

2b + 2h

Watch itpredict, then reveal

Example 25.1

AMC 8

A large square region is paved with $n^2$ gray square tiles, each measuring $s$ inches on a side. A border $d$ inches wide surrounds each tile. The figure below shows the case for $n = 3$ . When $n = 24$ , the $576$ gray tiles cover $64\%$ of the area of the large square region. What is the ratio $\frac{d}{s}$ for this larger value of $n$ ?

1
Walk across one row of the big square when $n = 24$ . You cross 24 tiles. How many borders of width $d$ do you cross?

Predict first:
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Contest cornerreal contest problems

Problem 25.7.1

AMC 8

Three identical rectangles are put together to form rectangle $ABCD$ , as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is 5 feet, what is the area in square feet of rectangle $ABCD$ ?

Problem 25.7.4

AMC 8

The midpoints of the four sides of a rectangle are $(-3, 0)$ , $(2, 0)$ , $(5, 4)$ , and $(0, 4)$ . What is the area of the rectangle?

Problem 25.7.5(additional)

AMC 8

Two congruent squares, $ABCD$ and $PQRS$ , have side length 15. They overlap to form the 15 by 25 rectangle $AQRD$ shown. What percent of the area of rectangle $AQRD$ is shaded?

Checkpoint for section 25.2

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

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Rhombus

In this section you will be able to find the area of a rhombus from its two diagonals.

A kite-shaped diamond on a playing card has all four sides equal. It is not a square, because it leans. How do you find its area?

A rhombus has 4 equal sides. Its diagonals are perpendicular and cut each other in half, so they split the rhombus into 4 matching right triangles.

Look:a rhombus fits inside a rectangle that is d₁ by d₂, and it fills exactly half of it.

Exploreplay first, formulas later

Rhombus

All 4 sides equal: yes
4 right angles: no
Both pairs of opposite sides parallel: yes
At least one pair of opposite sides parallel: yes
Diagonals are equal: no
Diagonals are perpendicular: yes
Diagonals cut each other in half: yes

Rhombus: area 24, perimeter 20

Try this: The diagonals d₁ and d₂ are drawn. Which checklist line tells you they cross at a right angle?
Try this: The diagonals cut the rhombus into 4 right triangles. Each has legs half of d₁ and half of d₂. Add up their areas. Do you get d₁ × d₂ ÷ 2?
Try this: Press Square. Is a square also a rhombus? Which checklist lines do both shapes tick?

Name it

Theorem 25.3.1 · Area of a Rhombus

A rhombus with diagonals $d_1$ and $d_2$ has an area of

\frac{1}{2}d_1 d_2

and a perimeter of

2 \times \sqrt{d_1^2 + d_2^2}

Checkpoint for section 25.3

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

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Parallelogram

In this section you will be able to find the area of a parallelogram with base × height.

Push the top of a rectangular box sideways and it leans into a parallelogram. The sides keep their length. Does the area stay the same?

A parallelogram has both pairs of opposite sides parallel. Cut a triangle off one end and slide it to the other end: you get a rectangle with the same base and the same height.

Look:the height is always measured straight up from the base, never along the slanted side.

Exploreplay first, formulas later

Parallelogram

All 4 sides equal: no
4 right angles: no
Both pairs of opposite sides parallel: yes
At least one pair of opposite sides parallel: yes
Diagonals are equal: no
Diagonals are perpendicular: no
Diagonals cut each other in half: yes

Parallelogram: area 30, perimeter ≈22.77

Try this: Turn on Height. Is the height the slanted side, or the straight-up distance between the bases?
Try this: Drag corners C and D the same number of steps to the right. The parallelogram leans more. Does the area change?
Try this: Press Rectangle. A rectangle is a parallelogram that stands up straight. Which checklist lines does it add?

Name it

Theorem 25.4.1 · Area of a Parallelogram

A parallelogram with base $b$ and height $h$ has an area of

bh

Checkpoint for section 25.4

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

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Trapezoid

In this section you will be able to find the area of a trapezoid by averaging its two bases.

A ramp is 8 feet wide at the bottom and 4 feet wide at the top. On average, it is 6 feet wide. Can you use that average to find its area?

A trapezoid has one pair of parallel sides, called the bases. Its area is the average of the two bases times the height between them.

Look:two right angles on the same side often hide a trapezoid. Two lines perpendicular to the same line are parallel.

Exploreplay first, formulas later

Try this: Turn on Height and Show the formula. Which two sides are the bases?
Try this: Drag corner C until the top base is as long as the bottom one. What shape do you get, and what happens to the formula?
Try this: Drag corner D down to the bottom row. Now one base has length 0. Which shape is that, and does the formula still work?

Name it

Theorem 25.5.1 · Area of a Trapezoid

A trapezoid with 2 bases $b_1$ and $b_2$ and a height $h$ has an area of

\frac{b_1 + b_2}{2} \cdot h

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Example 25.2

EMCC

In quadrilateral $PEAR$ , $PE = 21$ , $EA = 20$ , $AR = 15$ , $RE = 25$ , and $AP = 29$ . Find the area of the quadrilateral.

1
Look at triangle $PEA$ . Its sides are $21$ , $20$ and $29$ . What is $21^2 + 20^2$ ?

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Your prediction

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Contest cornerreal contest problems

Problem 25.7.3

AMC 8

Quadrilateral $ABCD$ is a trapezoid, $AD = 15$ , $AB = 50$ , $BC = 20$ , and the altitude is 12. What is the area of the trapezoid?

Checkpoint for section 25.5

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

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Breaking Quadrilaterals into Triangles

In this section you will be able to find the area of any quadrilateral by cutting it into triangles.

A strange four-sided field has no formula. But draw one line from corner to corner, and it becomes two triangles. You know how to do triangles!

Any quadrilateral can be cut along a diagonal into two triangles. You can also drop heights to make right triangles and rectangles. Add the pieces to get the whole area.

Look:two triangles with the same height have areas in the same ratio as their bases.

Exploreplay first, formulas later

Try this: Press Cut along AC, then Cut along BD. The two triangles change. Does the total change?
Try this: Press Drop the heights. What three pieces do you get? Add their areas.
Try this: Turn on Median. It joins the middles of the slanted sides. Is it the average of the two bases?

Watch itpredict, then reveal

Example 25.3

EMCC

Let $ABCD$ be a rectangle with $AB = 10$ and $BC = 12$ . Let $M$ be the midpoint of $CD$ , and $P$ be the point on $BM$ such that $BP = BC$ . Find the area of $ABPD$ .

1
Cut $ABPD$ along the diagonal $BD$ . What is the area of triangle $ABD$ ?

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Contest cornerreal contest problems

Problem 25.7.6(additional)

BmMT

Square $ABCD$ has side length 2. Square $ACEF$ is drawn such that $B$ lies inside square $ACEF$ . Compute the area of pentagon $AFECD$ ?

Problem 25.7.8(additional)

BmMT

Let $ABCD$ be a square of side length 10. Let $EFGH$ be a square of side length 15 such that $E$ is the center of $ABCD$ , $EF$ intersects $BC$ at $X$ , and $EH$ intersects $CD$ at $Y$ . If $BX = 7$ , what is the area of quadrilateral $EXCY$ ?

Checkpoint for section 25.6

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

25.7

Practice Problems

In this section you will be able to mix squares, rectangles and triangles on a tricky folding problem.

Grab a piece of paper and fold along! This last problem folds a square twice. Every problem has three hints if you get stuck.

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

Contest cornerreal contest problems

Problem 25.7.7(additional)

BmMT

Karthik has a paper square of side length 2. He folds the square along a crease that connects the midpoints of two opposite sides (as shown in the left diagram, where the dotted line indicates the fold). He takes the resulting rectangle and folds it such that one of its vertices lands on the vertex that is diagonally opposite. Find the area of Karthik's final figure.