Lesson 5.45.4 Small Group · Group 2
Start hereWords, worked example, and sentence starters
Learning target I can find the area of a composite figure by decomposing it into basic shapes, including a regular polygon split into triangles, and adding or subtracting the areas, explain why the method works, and use it on a problem I have not seen before.
Start hereWords for this lesson
Read each word before you begin. Say it out loud.
| Word | What it means | Example |
|---|---|---|
| Composite FigureSpanish: Figura compuesta | A shape made by putting two or more simple shapes together. | triangles joined together |
| DecomposeSpanish: Descomponer | To break a shape into smaller, simpler shapes. | hexagon → 6 triangles |
| AddSpanish: Sumar | Add up the areas of the smaller shapes to get the total. | T-shaped hallway: top rectangle = 30 sq ft, bottom rectangle = 28 sq ft → total = 30 + 28 = 58 sq ft |
| SubtractSpanish: Restar | Take away the area of a missing piece from a bigger shape. | A 14×10 pool with a 6×4 cutout: 140 − 24 = 116 sq ft of water surface |
| FormulaSpanish: Fórmula | A math rule written with symbols. | Rectangle: A = l × w; Triangle: A = ½ × b × h; use the right formula for each piece of a composite figure |
| Regular polygonSpanish: Polígono regular | A shape whose sides are all the same length and whose angles are all the same size. | stop sign, hexagon tile |
How it worksWorked examplearea of a regular polygon
These numbers are not on your problems. The steps are. Follow them with your own numbers.
A regular octagon has sides of 12 in. A triangle drawn from the center to one side has a height of 14 in. Find the area.
- A regular octagon has eight equal sides, so it decomposes into eight congruent triangles.
- Find the area of ONE triangle. ½ × 12 × 14 = 84
- Every triangle matches, so multiply by how many there are. 84 × 8 = 672
Answer: The area of the octagon is 672 square inches (672 in²).
Now you trySame steps, your turn
The steps are the same as the worked example. The numbers are yours. Do the work.
A regular hexagon has sides of 20 m. Each triangle drawn from the center has a height of 17 m. Find the area.
- How many congruent triangles does a regular hexagon decompose into?
- Area of ONE triangle. ½ × × =
- Multiply by the number of triangles. × =
Say it and write itSentence starters
Finish each sentence out loud with a partner. Then use them in your writing.
- I decomposed the figure into congruent .
- The area of one triangle is square units.
- There are triangles, so the total area is .
- I know the triangles are congruent because .
Word bank regular polygondecomposecongruenttrianglecenterbaseheightmultiplytotalareasquare unitscomposite figure
Watch outA common mistake
A common mistake in Area of Composite Figures is adding every piece even when a piece has been cut out of the figure. For a 10 ft by 8 ft rectangle with a 3 ft by 2 ft rectangular notch cut from one corner, students compute the full rectangle 10 × 8 = 80 sq ft and stop, or they add the notch instead of removing it (80 + 6 = 86 sq ft) instead of subtracting it: 80 − (3 × 2) = 80 − 6 = 74 sq ft. Before submitting, check whether each piece is part of the figure (add it) or missing from the figure (subtract it).
Lesson 5.45.4 Small Group · Group 2
Group 2Challenge practice
Learning target I can find the area of a composite figure by decomposing it into basic shapes, including a regular polygon split into triangles, and adding or subtracting the areas, explain why the method works, and use it on a problem I have not seen before.
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1Complete each step, then write the final answer.
Two sides are missing. The dashed corner is outside the room. An L-shaped room has a bottom side of 12 m and a left side of 9 m. The top side is 5 m. The short right side is 4 m. Find the area.
- 1Find the two missing side lengths.
- 2Split into two rectangles and add. What is the area?
- 3Check: subtract the missing corner from the big rectangle.
Final answer: -
2Complete each step, then write the final answer.
The dashed triangle is cut out. A rectangular sign is 10 in by 6 in. A triangle is cut out of it. The triangle has a base of 4 in and a height of 3 in. Find the area of the sign that is left.
- 1Find the area of the whole rectangle.
- 2Find the triangle. Then find the area that is left.
- 3Why do we subtract the triangle and not add it?
Final answer: -
3Complete each step, then write the final answer.
A regular polygon splits into congruent triangles that meet at the center. Each triangle has a base of 5 cm and a height of 6 cm. The polygon’s area is 120 square centimeters. How many sides does it have?
Show your workFinal answer: -
4Complete each step, then write the final answer.
An L-shaped garden has a bottom side of 14 ft and a left side of 10 ft. The top side is 8 ft. The short right side is 6 ft. Find the area.
Show your workFinal answer: -
5Complete each step, then write the final answer.
A rectangular lawn is 25 m by 18 m. A flower bed inside it is a trapezoid with bases 4 m and 8 m and height 5 m. How many square meters of the lawn are grass?
Show your workFinal answer: -
6Complete each step, then write the final answer.
An L-shape fits in a 10 ft by 8 ft rectangle with a 4 ft by 4 ft corner missing. Ana adds two pieces and gets 58 square feet. Ben gets 64 square feet. What is the correct area?
Show your workFinal answer: