Level 3 · Challenge ← Back to lesson

Composite and Shaded Figures: Practice

12 multiple-choice questions, progressively harder.

Level 3 · Challenge 0 / 12 answered

Question 1 of 12

1

A rectangle has width $w$ cm and height $9$ cm. A triangle outside it is joined along the full side of width $w$ cm and has perpendicular height $4$ cm. The union area is $143$ square centimeters. Find $w$ .

Answer choices for question 1 A $13$ cm B $11$ cm C $\frac{143}{9}$ cm D $22$ cm
2

Compare two separate $31$ cm by $20$ cm rectangles. From each, remove a $4$ cm wide and $6$ cm deep rectangle. In Case A it is a corner notch; in Case B it opens into the middle of the top edge. How much larger is the perimeter in Case B?

Answer choices for question 2 A $0$ cm B $8$ cm C $12$ cm D $20$ cm
3

Squares of side lengths $11$ cm and $9$ cm have disjoint interiors and meet along exactly one segment of length $7$ cm. Find their union area and perimeter.

Answer choices for question 3 A
$153\text{ cm}^2$

$66$ cm
B
$202\text{ cm}^2$

$66$ cm
C
$202\text{ cm}^2$

$73$ cm
D
$202\text{ cm}^2$

$80$ cm
4

A $30$ cm by $24$ cm rectangle contains a rectangular hole of width $x$ cm and height $10$ cm strictly inside. The retained area is $590$ square centimeters. What is the total boundary length including the hole?

Answer choices for question 4 A $108$ cm B $134$ cm C $62$ cm D $154$ cm
5

A $32$ cm by $21$ cm rectangle loses a right-triangular corner with perpendicular legs $9$ cm and $12$ cm along the rectangle sides. Find the retained area and perimeter.

Answer choices for question 5 A
$564\text{ cm}^2$

$100$ cm
B
$618\text{ cm}^2$

$100$ cm
C
$618\text{ cm}^2$

$106$ cm
D
$618\text{ cm}^2$

$121$ cm
6

Two polygonal regions have areas $132$ and $90$ square centimeters. Their union has area $190$ square centimeters. What is the area common to both?

Answer choices for question 6 A $42\text{ cm}^2$ B $58\text{ cm}^2$ C $32\text{ cm}^2$ D $100\text{ cm}^2$
7

Two disjoint polygonal regions have areas $350$ square millimeters and $0.018$ square meters. Use $1$ cm equal to $10$ mm and $1$ m equal to $100$ cm. What is their combined area in square centimeters?

Answer choices for question 7 A $183.5\text{ cm}^2$ B $36.8\text{ cm}^2$ C $530\text{ cm}^2$ D $5.3\text{ cm}^2$
8

A $28$ cm by $19$ cm rectangle has a rectangular notch $12$ cm wide and $8$ cm deep opening into the middle of the top edge. What is the remaining polygon's perimeter?

Answer choices for question 8 A $94$ cm B $134$ cm C $118$ cm D $110$ cm
9

A trapezoid has parallel bases $26$ cm and $14$ cm and perpendicular height $15$ cm. A rectangular strip of height $15$ cm lies entirely within the trapezoid and spans between its parallel bases. Removing the strip leaves $225$ square centimeters. What is the strip width?

Answer choices for question 9 A $15$ cm B $10$ cm C $5$ cm D $20$ cm
10

A $19$ cm by $11$ cm rectangle contains a rectangular hole strictly inside. The hole is $40$ mm high, and the retained area is $193$ square centimeters. Use $10$ mm equal to $1$ cm and $100$ cm equal to $1$ m. What is the total boundary length, including the hole, in meters?

Answer choices for question 10 A $0.76$ m B $0.60$ m C $7.6$ m D $0.68$ m
11

A $33$ cm by $17$ cm rectangle contains two holes strictly inside it with disjoint interiors. One is a right triangle with perpendicular legs $8$ cm and $9$ cm. The other is a $7$ cm by $4$ cm rectangle. What area remains?

Answer choices for question 11 A $461\text{ cm}^2$ B $525\text{ cm}^2$ C $553\text{ cm}^2$ D $497\text{ cm}^2$
12

A simple polygon is decomposed in two different ways. Each decomposition uses pieces with disjoint interiors and covers exactly the polygon. What must be true?

Answer choices for question 12 A The sums of the piece perimeters are equal. B The sums of the piece areas are equal. C Both decompositions have the same number of pieces. D Every piece in one decomposition is congruent to a piece in the other.

← Back to lesson