Middle School Math Quiz: Find Area By Composing And Decomposing

20 questions · exam conditions

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Find Area By Composing And DecomposingQuestion 1 of 20

Look at the composite figure. To find its area, Jason decomposes it into simpler shapes. Which decomposition strategy will give him the correct total area?

$Question graphic$

Two rectangles:

(6 \times 4) + (8 \times 3) = 48 \text{ square units}

One large rectangle minus one small rectangle:

(8 \times 7) - (2 \times 3) = 50 \text{ square units}

Two rectangles:

(6 \times 4) + (2 \times 3) = 30 \text{ square units}

One trapezoid with parallel sides 6 and 8, height 4:

\frac{1}{2}(6 + 8) \times 4 = 28 \text{ square units}

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Middle School Math Quiz

Middle School Math Quiz: Find Area By Composing And Decomposing

Practice Find Area By Composing And Decomposing in Middle School Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

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This quiz focuses on Find Area By Composing And Decomposing, giving you a quick way to practice the rules, question types, and explanations that matter most for Middle School Math.

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Question 1

Look at the composite figure. To find its area, Jason decomposes it into simpler shapes. Which decomposition strategy will give him the correct total area?

Two rectangles: $(6 \times 4) + (8 \times 3) = 48 \text{ square units}$
One large rectangle minus one small rectangle: $(8 \times 7) - (2 \times 3) = 50 \text{ square units}$ (correct answer)
Two rectangles: $(6 \times 4) + (2 \times 3) = 30 \text{ square units}$
One trapezoid with parallel sides 6 and 8, height 4: $\frac{1}{2}(6 + 8) \times 4 = 28 \text{ square units}$

Explanation: The L-shaped figure can be decomposed as a large 8×7 rectangle (56 sq units) minus the missing 2×3 rectangle in the upper right (6 sq units), giving 50 sq units. Choice A incorrectly identifies the dimensions of the right rectangle. Choice C uses wrong dimensions for both rectangles. Choice D incorrectly treats the figure as a trapezoid, which it is not.

Question 2

A table top is shaped like a trapezoid with parallel sides $6\text{ cm}$ and $10\text{ cm}$ and height $4\text{ cm}$ . What is the area of the trapezoid?

$16\text{ cm}^2$
$32\text{ cm}^2$ (correct answer)
$64\text{ cm}^2$
$24\text{ cm}^2$

Question 3

Which method correctly finds the area of an L-shaped floor made from a large $8\text{ m} \times 6\text{ m}$ rectangle with a $3\text{ m} \times 2\text{ m}$ corner cut out?

Subtract: $8\cdot 6 - 3\cdot 2$ (correct answer)
Subtract: $3\cdot 2 - 8\cdot 6$
Add: $8\cdot 6 + 3\cdot 2$
Multiply: $(8+3)(6+2)$

Question 4

A school garden is shaped like a right triangle with base $6\text{ m}$ and height $4\text{ m}$ . What is the area of the garden?

$24\text{ m}^2$
$12\text{ m}^2$ (correct answer)
$10\text{ m}^2$
$28\text{ m}^2$

Question 5

Which strategy correctly finds the area of an L-shaped floor that fits inside an $11\text{ ft}\times 9\text{ ft}$ rectangle with a $4\text{ ft}\times 3\text{ ft}$ rectangular corner cut out?

Subtract: $(4\times 3)-(11\times 9)$
Multiply: $11\times 9\times 4\times 3$
Add: $(11\times 9)+(4\times 3)$
Subtract: $(11\times 9)-(4\times 3)$ (correct answer)

Explanation: This question tests finding the area of polygons by composing into rectangles (large minus cutout) or decomposing into triangles/rectangles (sum parts), using A=(1/2)bh for triangles, A=lw for rectangles. Strategies: decompose (break into simpler shapes: triangle as half rectangle with A=(1/2)bh, L-shape as two rectangles summing areas), compose (enclose in rectangle, subtract outside parts: right triangle in 6×4 rectangle is (1/2)×24=12). Formulas: rectangle A=lw (length×width), triangle A=(1/2)bh (base×height divided by 2—half of rectangle). Example: trapezoid bases 6 and 10, height 4, decompose into rectangle 6×4=24 plus triangle with base 4, height 4: (1/2)×4×4=8, total 24+8=32. Example: right triangle base 6, height 4, fits in rectangle 6×4=24, triangle is half: (1/2)×24=12, or directly (1/2)×6×4=12; or L-shape: large rectangle 8×6=48 minus cutout 3×2=6 gives 48-6=42, or decompose into rectangles 5×6=30 and 3×4=12, sum 30+12=42 (both methods work). The correct strategy is subtracting the cutout from the large rectangle: (11 × 9) - (4 × 3) = 99 - 12 = 87 ft². A common error is adding the cutout instead ((11 × 9) + (4 × 3) = 111 ft²), or subtracting in the wrong order ((4 × 3) - (11 × 9) = -87 ft²), or multiplying all dimensions. Decomposing: (1) identify simpler shapes (can polygon be split into rectangles and triangles?), (2) draw lines dividing (dotted lines showing decomposition), (3) calculate each part (rectangle: lw, triangle: (1/2)bh), (4) sum (total=part₁+part₂+...). Composing: (1) enclose in rectangle (smallest rectangle containing polygon), (2) identify cutouts (triangles or rectangles outside polygon), (3) calculate rectangle and cutouts, (4) subtract (rectangle area minus cutout areas). Triangle key: A=(1/2)bh always (half of rectangle with same base and height). Real-world: room area for flooring (L-shaped room composed/decomposed), garden area for fencing, window area for glass. Mistakes: forgetting (1/2) for triangles, wrong decomposition (incorrect shapes), arithmetic errors, units not squared.

Question 6

A right triangle banner has a base of $6\text{ ft}$ and a height of $4\text{ ft}$ . What is its area?

$10\text{ ft}^2$
$24\text{ ft}^2$
$12\text{ ft}^2$ (correct answer)
$20\text{ ft}^2$

Explanation: This question tests finding the area of polygons by composing into rectangles (large minus cutout) or decomposing into triangles/rectangles (sum parts), using A=(1/2)bh for triangles, A=lw for rectangles. Strategies: decompose (break into simpler shapes: triangle as half rectangle with A=(1/2)bh, L-shape as two rectangles summing areas), compose (enclose in rectangle, subtract outside parts: right triangle in 6×4 rectangle is (1/2)×24=12). Formulas: rectangle A=lw (length×width), triangle A=(1/2)bh (base×height divided by 2—half of rectangle). Example: trapezoid bases 6 and 10, height 4, decompose into rectangle 6×4=24 plus triangle with base 4, height 4: (1/2)×4×4=8, total 24+8=32. Example: right triangle base 6, height 4, fits in rectangle 6×4=24, triangle is half: (1/2)×24=12, or directly (1/2)×6×4=12; or L-shape: large rectangle 8×6=48 minus cutout 3×2=6 gives 48-6=42, or decompose into rectangles 5×6=30 and 3×4=12, sum 30+12=42 (both methods work). The correct area is found using the triangle formula: (1/2) × 6 ft × 4 ft = 12 ft², or by composing into a 6 ft × 4 ft rectangle (24 ft²) and taking half. A common error is forgetting the 1/2 and calculating as a rectangle (6 × 4 = 24 ft²), or mixing up base and height, or arithmetic like (1/2) × 6 × 4 = 24 instead of 12. Decomposing: (1) identify simpler shapes (can polygon be split into rectangles and triangles?), (2) draw lines dividing (dotted lines showing decomposition), (3) calculate each part (rectangle: lw, triangle: (1/2)bh), (4) sum (total=part₁+part₂+...). Composing: (1) enclose in rectangle (smallest rectangle containing polygon), (2) identify cutouts (triangles or rectangles outside polygon), (3) calculate rectangle and cutouts, (4) subtract (rectangle area minus cutout areas). Triangle key: A=(1/2)bh always (half of rectangle with same base and height). Real-world: room area for flooring (L-shaped room composed/decomposed), garden area for fencing, window area for glass. Mistakes: forgetting (1/2) for triangles, wrong decomposition (incorrect shapes), arithmetic errors, units not squared.

Question 7

An irregular patio can be decomposed into two rectangles that do not overlap: Rectangle 1 is $5\text{ m} \times 6\text{ m}$ and Rectangle 2 is $3\text{ m} \times 4\text{ m}$ . What is the total area of the patio?

$34\text{ m}^2$
$42\text{ m}^2$ (correct answer)
$46\text{ m}^2$
$54\text{ m}^2$

Question 8

A triangular pennant is a right triangle with base $6\text{ cm}$ and height $4\text{ cm}$ . What is its area? (Use $A=\tfrac{1}{2}bh$ .)

$12\text{ cm}^2$ (correct answer)
$10\text{ cm}^2$
$20\text{ cm}^2$
$24\text{ cm}^2$

Explanation: This question tests finding the area of a right triangle by using A = (1/2)bh, composing it as half of a rectangle. Strategies include directly applying the formula for the triangle with base 6 cm and height 4 cm, or composing a 6×4 rectangle of 24 cm² and taking half for 12 cm². Formulas: triangle A = (1/2)bh, which is half of a rectangle's A = lw with same base and height; example, trapezoid bases 6 and 10, height 4, decomposes to rectangle 24 plus triangle 8 for 32. For example, this right triangle base 6, height 4 is (1/2)×6×4=12, or half of 24=12; an L-shape might be 48 minus 6=42 by composition. The correct area is (1/2)×6×4 = 12 cm² using the formula. A common error is forgetting the 1/2, calculating bh=24 instead of 12, or mixing with rectangle formula. For triangles: always use (1/2)bh; real-world like pennant fabric area. Steps: identify base and height, multiply and halve; mistakes include arithmetic errors or unsquared units.

Question 9

A trapezoid-shaped window has parallel sides $6\text{ in}$ and $10\text{ in}$ and height $4\text{ in}$ . A student computes $(6+10)\cdot 4=64\text{ in}^2$ . What is the correct area?

$32\text{ in}^2$ (correct answer)
$40\text{ in}^2$
$16\text{ in}^2$
$64\text{ in}^2$

Question 10

A tabletop is shaped like a trapezoid. The parallel sides are $6\text{ ft}$ and $10\text{ ft}$ , and the height is $4\text{ ft}$ . What is the area of the tabletop? (You may use $A=\tfrac{1}{2}(b_1+b_2)h$ .)

$64\text{ ft}^2$
$32\text{ ft}^2$ (correct answer)
$16\text{ ft}^2$
$40\text{ ft}^2$

Explanation: This question tests finding the area of a trapezoid using A = (1/2)(b1 + b2)h, which relates to decomposing into a rectangle and triangles. Strategies include using the formula directly for bases 6 ft and 10 ft, height 4 ft, giving (1/2)×16×4=32 ft², or decomposing into a rectangle and triangle. Formulas: trapezoid as average bases times height, rectangle A=lw, triangle (1/2)bh; example, this trapezoid decomposes to 6×4=24 plus (1/2)×4×4=8 for 32. Like a right triangle in 6×4=24, half is 12; or L-shape 48-6=42. The correct area is (1/2)(6+10)×4 = 32 ft² using the formula. Errors include not halving, getting 16×4=64, or adding bases wrong like 6+10=18. Decomposing: (1) split into rectangle and triangle, (2) calculate each, (3) sum; real-world for tabletops like painting area. Mistakes: forgetting 1/2 in formula, arithmetic like 8×4=36 instead of 32, squared units.

Question 11

A storage mat is an irregular shape that can be found by composing a large rectangle and subtracting a triangular cutout.

The large rectangle is $12\text{ ft}$ by $7\text{ ft}$ .
A right-triangular cutout has base $6\text{ ft}$ and height $3\text{ ft}$ .

What is the area of the mat?

$66\text{ ft}^2$
$93\text{ ft}^2$
$84\text{ ft}^2$
$75\text{ ft}^2$ (correct answer)

Explanation: This question tests finding the area of an irregular mat by composing a large rectangle and subtracting a triangular cutout, using A = lw minus (1/2)bh. Strategies: large 12 ft × 7 ft = 84 ft², cutout (1/2)×6×3=9 ft², area 84-9=75 ft². Formulas: rectangle A = lw, triangle (1/2)bh; example, trapezoid 24 + 8 = 32. Like L-shape 48-6=42, triangle 12. The correct area is 84 - 9 = 75 ft² using composition. Errors include full triangle 18, 84-18=66, or no subtract 84. Composing: (1) enclose, (2) identify cutout, (3) calculate, (4) subtract; real-world for mats like cleaning area. Mistakes: forgetting 1/2, arithmetic like 84-9=74, unsquared units.

Question 12

A stage platform is an L-shape that can be decomposed into two rectangles:

Rectangle 1: $5\text{ m} \times 6\text{ m}$
Rectangle 2: $3\text{ m} \times 4\text{ m}$

What is the total area of the platform?

$42\text{ m}^2$ (correct answer)
$12\text{ m}^2$
$30\text{ m}^2$
$48\text{ m}^2$

Explanation: This question tests finding the area of an L-shaped platform by decomposing into two rectangles and summing, using A = lw for each. Strategies: decompose into 5 m × 6 m = 30 m² and 3 m × 4 m = 12 m², total 42 m², or compose and subtract. Formulas: rectangle A = lw; triangle (1/2)bh; example, trapezoid 24 + 8 = 32. Like L-shape compose 48 - 6 = 42, or triangle half of 24 = 12. The correct area is 30 + 12 = 42 m² using decomposition. Errors include only one rectangle like 30, or multiplying wrong like 5×6=35. Decomposing: (1) identify rectangles, (2) calculate each, (3) sum; composing: enclose and subtract. Real-world for stages like carpeting; mistakes: overlap addition, arithmetic, unsquared units.

Question 13

A student says the area of a right triangle with base $6\text{ cm}$ and height $4\text{ cm}$ is $24\text{ cm}^2$ because $6\times 4=24$ . Which statement best checks the student's answer?

The student is correct because $A=bh$ for triangles.
The student forgot to square the units; the area should be $24\text{ cm}$ .
The student should add the base and height; the area is $6+4=10\text{ cm}^2$ .
The student should divide by 2; the area is $\tfrac{1}{2}\cdot 6\cdot 4=12\text{ cm}^2$ . (correct answer)

Question 14

A banner is a triangle with base $14\text{ in}$ and height $6\text{ in}$ . A student says the area is $84\text{ in}^2$ .

Is the student correct?

No, because $A=\tfrac{1}{2}bh=\tfrac{1}{2}\times 14\times 6=42\text{ in}^2$ . (correct answer)
Yes, because $A=\tfrac{1}{2}bh=\tfrac{1}{2}\times 14\times 6=84\text{ in}^2$ .
Yes, because $A=bh=14\times 6=84\text{ in}^2$ .
No, because $A=bh=14+6=20\text{ in}^2$ .

Explanation: This question tests verifying the area of a triangle using A = (1/2)bh, checking if the student's 84 in² is correct for base 14 in, height 6 in. Strategies: apply formula (1/2)×14×6=42 in², so student is wrong by not halving. Formulas: triangle (1/2)bh, half of rectangle bh; example, trapezoid 24 + 8 = 32. Like triangle (1/2)×6×4=12, not 24; L-shape correct 42. The student is incorrect because the area is 42 in², not 84. Errors include using bh=84 without 1/2, or adding like 14+6=20. For triangles: identify base/height, halve product; real-world for banners like material. Mistakes: forgetting 1/2, using perimeter, arithmetic, no units.

Question 15

A playground area is a pentagon that can be decomposed into a rectangle and a triangle. The rectangle is $8\text{ yd} \times 5\text{ yd}$ , and attached to one side is a triangle with base $8\text{ yd}$ and height $3\text{ yd}$ . What is the total area of the playground?

$28\text{ yd}^2$
$64\text{ yd}^2$
$40\text{ yd}^2$
$52\text{ yd}^2$ (correct answer)

Question 16

The figure shows a hexagon that can be divided into rectangles and triangles. Sarah calculates the area by decomposing it into two rectangles and two triangles. If her rectangles have areas of 35 and 42 square centimeters, and her triangles have areas of 15 and 18 square centimeters, what is the total area of the hexagon?

$95 \text{ square centimeters}$
$110 \text{ square centimeters}$ (correct answer)
$125 \text{ square centimeters}$
$140 \text{ square centimeters}$

Explanation: To find the total area, add all the component areas: 35 + 42 + 15 + 18 = 110 square centimeters. Choice A omits one triangle (110 - 15 = 95). Choice C adds an extra 15 (possibly double-counting one triangle). Choice D appears to use incorrect component areas or double-counts multiple shapes.

Question 17

Maya is designing a garden bed in the shape shown in the figure. She wants to plant flowers in the triangular sections and vegetables in the rectangular sections. What is the total area available for planting vegetables?

$84 \text{ square feet}$ (correct answer)
$96 \text{ square feet}$
$108 \text{ square feet}$
$120 \text{ square feet}$

Explanation: The figure can be decomposed into a large rectangle (12 × 10 = 120 sq ft) minus two triangles. The triangles each have base 6 ft and height 6 ft, so each triangle has area ½ × 6 × 6 = 18 sq ft. Total triangular area = 36 sq ft. Rectangular area for vegetables = 120 - 36 = 84 sq ft. Choice B incorrectly calculates one triangle as 24 sq ft. Choice C uses the wrong base measurement. Choice D fails to subtract the triangular areas.

Question 18

A stage backdrop is shaped like a rectangle with a triangular piece cut out. The full rectangle is $10\text{ ft} \times 6\text{ ft}$ . The cut-out is a right triangle with base $4\text{ ft}$ and height $3\text{ ft}$ . What is the area of the remaining backdrop?

$54\text{ ft}^2$ (correct answer)
$48\text{ ft}^2$
$66\text{ ft}^2$
$60\text{ ft}^2$

Question 19

A craftsman is making a decorative wooden piece by combining shapes. He starts with a square measuring 16 cm on each side, then adds four identical right triangles to the outside of each edge. Each triangle has legs of 8 cm and 6 cm. What is the total area of the completed decorative piece?

$256 \text{ square centimeters}$
$352 \text{ square centimeters}$ (correct answer)
$448 \text{ square centimeters}$
$512 \text{ square centimeters}$

Explanation: Square area: 16² = 256 sq cm. Each triangle area: ½ × 8 × 6 = 24 sq cm. Four triangles: 4 × 24 = 96 sq cm. Total area: 256 + 96 = 352 sq cm. Choice A only includes the square. Choice C incorrectly calculates each triangle as 48 sq cm (using 8 × 6 instead of ½ × 8 × 6). Choice D doubles the square area.

Question 20

A pentagon-shaped sign can be decomposed into a rectangle and a triangle that sit on top of it.

The rectangle is $8\text{ ft}$ wide and $5\text{ ft}$ tall.
The triangle on top has the same base as the rectangle ( $8\text{ ft}$ ) and a height of $3\text{ ft}$ .

What is the area of the pentagon?

$28\text{ ft}^2$
$40\text{ ft}^2$
$52\text{ ft}^2$ (correct answer)
$64\text{ ft}^2$

Explanation: This question tests finding the area of polygons by composing into rectangles (large minus cutout) or decomposing into triangles/rectangles (sum parts), using A=(1/2)bh for triangles, A=lw for rectangles. Strategies: decompose (break into simpler shapes: triangle as half rectangle with A=(1/2)bh, L-shape as two rectangles summing areas), compose (enclose in rectangle, subtract outside parts: right triangle in 6×4 rectangle is (1/2)×24=12). Formulas: rectangle A=lw (length×width), triangle A=(1/2)bh (base×height divided by 2—half of rectangle). Example: trapezoid bases 6 and 10, height 4, decompose into rectangle 6×4=24 plus triangle with base 4, height 4: (1/2)×4×4=8, total 24+8=32. Example: right triangle base 6, height 4, fits in rectangle 6×4=24, triangle is half: (1/2)×24=12, or directly (1/2)×6×4=12; or L-shape: large rectangle 8×6=48 minus cutout 3×2=6 gives 48-6=42, or decompose into rectangles 5×6=30 and 3×4=12, sum 30+12=42 (both methods work). The correct area is found by decomposing: rectangle 8 ft × 5 ft = 40 ft² plus triangle (1/2) × 8 ft × 3 ft = 12 ft², total 52 ft². A common error is forgetting the 1/2 for the triangle (40 + 24 = 64 ft²), or using rectangle height for triangle base, or arithmetic like 40 + 12 = 54. Decomposing: (1) identify simpler shapes (can polygon be split into rectangles and triangles?), (2) draw lines dividing (dotted lines showing decomposition), (3) calculate each part (rectangle: lw, triangle: (1/2)bh), (4) sum (total=part₁+part₂+...). Composing: (1) enclose in rectangle (smallest rectangle containing polygon), (2) identify cutouts (triangles or rectangles outside polygon), (3) calculate rectangle and cutouts, (4) subtract (rectangle area minus cutout areas). Triangle key: A=(1/2)bh always (half of rectangle with same base and height). Real-world: room area for flooring (L-shaped room composed/decomposed), garden area for fencing, window area for glass. Mistakes: forgetting (1/2) for triangles, wrong decomposition (incorrect shapes), arithmetic errors, units not squared.