Convex and Concave Polygons Lesson Plans & Worksheets - Free Printable
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Step-by-step solution for: Convex and Concave Polygons Lesson Plans & Worksheets
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Step-by-step solution for: Convex and Concave Polygons Lesson Plans & Worksheets
Here are the step-by-step solutions for the problems on the worksheet.
1, 2, and 3: Identify Polygons
* Definition: A polygon is a closed shape made of straight line segments. It cannot have curves, and the lines must connect end-to-end without crossing over each other in a messy way (simple polygon).
* Figure 1: This is a closed shape with straight sides. Count the sides: there are 8 sides.
* Answer: Polygon; Octagon.
* Figure 2: This shape has a curved bottom. Polygons must have only straight sides.
* Answer: Not a polygon.
* Figure 3: The lines cross each other and do not form a single closed loop of non-intersecting segments.
* Answer: Not a polygon.
4. Equiangular but not Equilateral
* Concept: "Equiangular" means all angles are equal. "Equilateral" means all sides are equal. A regular polygon has both. We need a shape where angles are the same, but sides can be different lengths.
* Reasoning: Think of a rectangle. All four corners are $90^\circ$ (equiangular), but the top/bottom can be long while the sides are short (not equilateral). A square is a special rectangle where sides are equal, but a general rectangle is not.
* Answer: Rectangle.
5, 6, and 7: Regular/Irregular and Concave/Convex
* Definitions:
* Regular: All sides equal AND all angles equal.
* Irregular: Sides or angles are not all equal.
* Convex: No part of the shape points inward (like a standard stop sign). All interior angles are less than $180^\circ$.
* Concave: At least one part points inward (like a cave or a star). At least one interior angle is greater than $180^\circ$.
* Figure 5: The shape has "dents" pointing inward. The tick marks show sides are equal, but the angles are clearly different (some pointy, some wide).
* Answer: Irregular, Concave.
* Figure 6: The tick marks on all sides mean they are equal length. The arc marks on all angles mean they are equal measure. It bulges outward everywhere.
* Answer: Regular, Convex.
* Figure 7: The tick marks show sides are equal. However, look at the angles: some have double arcs, some have single arcs. Since the angles are not all the same, it is not regular. It bulges outward.
* Answer: Irregular, Convex.
8. Sum of Interior Angles of a 14-gon
* Formula: Sum $= (n - 2) \times 180^\circ$, where $n$ is the number of sides.
* Calculation:
* $n = 14$
* $14 - 2 = 12$
* $12 \times 180 = 2160$
* Answer: $2160^\circ$.
9. Measure of Each Interior Angle of Hexagon ABCDEF
* Method 1 (Using the diagram): The diagram labels every angle as $5a^\circ$. Since it is a hexagon ($n=6$), we can find the total sum first.
* Sum $= (6 - 2) \times 180 = 4 \times 180 = 720^\circ$.
* There are 6 angles, so each angle is $720 / 6 = 120^\circ$.
* Method 2 (Solving for 'a'):
* $6 \times 5a = 720$
* $30a = 720$
* $a = 24$
* Angle $= 5a = 5(24) = 120^\circ$.
* Answer: $120^\circ$.
10. Value of n in Pentagon PQRST
* Concept: The sum of exterior angles of any convex polygon is always $360^\circ$.
* Equation: Add up the given exterior angles and set them equal to 360.
* Angles given: $60^\circ, 50^\circ, 2n^\circ, 70^\circ, n^\circ$.
* $60 + 50 + 2n + 70 + n = 360$
* Calculation:
* Combine numbers: $60 + 50 + 70 = 180$.
* Combine variables: $2n + n = 3n$.
* $180 + 3n = 360$
* Subtract 180 from both sides: $3n = 180$.
* Divide by 3: $n = 60$.
* Answer: $60$.
11. Interior Angle of the Waterwheel (Regular Octagon)
* Shape: Regular Octagon ($n = 8$).
* Formula for one interior angle: $\frac{(n - 2) \times 180}{n}$
* Calculation:
* Sum of angles $= (8 - 2) \times 180 = 6 \times 180 = 1080^\circ$.
* One angle $= 1080 / 8$.
* $1080 / 8 = 135$.
* Answer: $135^\circ$.
12. Exterior Angle of the Waterwheel (Regular Octagon)
* Concept: The sum of exterior angles is always $360^\circ$. For a *regular* polygon, you just divide by the number of sides. Alternatively, Interior + Exterior = $180^\circ$.
* Calculation:
* Method 1: $360 / 8 = 45^\circ$.
* Method 2: $180 - 135 (\text{interior}) = 45^\circ$.
* Answer: $45^\circ$.
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Final Answer:
1. Polygon; Octagon
2. Not a polygon
3. Not a polygon
4. Rectangle
5. Irregular, Concave
6. Regular, Convex
7. Irregular, Convex
8. $2160^\circ$
9. $120^\circ$
10. $60$
11. $135^\circ$
12. $45^\circ$
1, 2, and 3: Identify Polygons
* Definition: A polygon is a closed shape made of straight line segments. It cannot have curves, and the lines must connect end-to-end without crossing over each other in a messy way (simple polygon).
* Figure 1: This is a closed shape with straight sides. Count the sides: there are 8 sides.
* Answer: Polygon; Octagon.
* Figure 2: This shape has a curved bottom. Polygons must have only straight sides.
* Answer: Not a polygon.
* Figure 3: The lines cross each other and do not form a single closed loop of non-intersecting segments.
* Answer: Not a polygon.
4. Equiangular but not Equilateral
* Concept: "Equiangular" means all angles are equal. "Equilateral" means all sides are equal. A regular polygon has both. We need a shape where angles are the same, but sides can be different lengths.
* Reasoning: Think of a rectangle. All four corners are $90^\circ$ (equiangular), but the top/bottom can be long while the sides are short (not equilateral). A square is a special rectangle where sides are equal, but a general rectangle is not.
* Answer: Rectangle.
5, 6, and 7: Regular/Irregular and Concave/Convex
* Definitions:
* Regular: All sides equal AND all angles equal.
* Irregular: Sides or angles are not all equal.
* Convex: No part of the shape points inward (like a standard stop sign). All interior angles are less than $180^\circ$.
* Concave: At least one part points inward (like a cave or a star). At least one interior angle is greater than $180^\circ$.
* Figure 5: The shape has "dents" pointing inward. The tick marks show sides are equal, but the angles are clearly different (some pointy, some wide).
* Answer: Irregular, Concave.
* Figure 6: The tick marks on all sides mean they are equal length. The arc marks on all angles mean they are equal measure. It bulges outward everywhere.
* Answer: Regular, Convex.
* Figure 7: The tick marks show sides are equal. However, look at the angles: some have double arcs, some have single arcs. Since the angles are not all the same, it is not regular. It bulges outward.
* Answer: Irregular, Convex.
8. Sum of Interior Angles of a 14-gon
* Formula: Sum $= (n - 2) \times 180^\circ$, where $n$ is the number of sides.
* Calculation:
* $n = 14$
* $14 - 2 = 12$
* $12 \times 180 = 2160$
* Answer: $2160^\circ$.
9. Measure of Each Interior Angle of Hexagon ABCDEF
* Method 1 (Using the diagram): The diagram labels every angle as $5a^\circ$. Since it is a hexagon ($n=6$), we can find the total sum first.
* Sum $= (6 - 2) \times 180 = 4 \times 180 = 720^\circ$.
* There are 6 angles, so each angle is $720 / 6 = 120^\circ$.
* Method 2 (Solving for 'a'):
* $6 \times 5a = 720$
* $30a = 720$
* $a = 24$
* Angle $= 5a = 5(24) = 120^\circ$.
* Answer: $120^\circ$.
10. Value of n in Pentagon PQRST
* Concept: The sum of exterior angles of any convex polygon is always $360^\circ$.
* Equation: Add up the given exterior angles and set them equal to 360.
* Angles given: $60^\circ, 50^\circ, 2n^\circ, 70^\circ, n^\circ$.
* $60 + 50 + 2n + 70 + n = 360$
* Calculation:
* Combine numbers: $60 + 50 + 70 = 180$.
* Combine variables: $2n + n = 3n$.
* $180 + 3n = 360$
* Subtract 180 from both sides: $3n = 180$.
* Divide by 3: $n = 60$.
* Answer: $60$.
11. Interior Angle of the Waterwheel (Regular Octagon)
* Shape: Regular Octagon ($n = 8$).
* Formula for one interior angle: $\frac{(n - 2) \times 180}{n}$
* Calculation:
* Sum of angles $= (8 - 2) \times 180 = 6 \times 180 = 1080^\circ$.
* One angle $= 1080 / 8$.
* $1080 / 8 = 135$.
* Answer: $135^\circ$.
12. Exterior Angle of the Waterwheel (Regular Octagon)
* Concept: The sum of exterior angles is always $360^\circ$. For a *regular* polygon, you just divide by the number of sides. Alternatively, Interior + Exterior = $180^\circ$.
* Calculation:
* Method 1: $360 / 8 = 45^\circ$.
* Method 2: $180 - 135 (\text{interior}) = 45^\circ$.
* Answer: $45^\circ$.
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Final Answer:
1. Polygon; Octagon
2. Not a polygon
3. Not a polygon
4. Rectangle
5. Irregular, Concave
6. Regular, Convex
7. Irregular, Convex
8. $2160^\circ$
9. $120^\circ$
10. $60$
11. $135^\circ$
12. $45^\circ$
Parent Tip: Review the logic above to help your child master the concept of polygon attributes worksheet.
