Math worksheet for calculating interior angles of polygons, featuring solved examples and practice problems.
Interior Angles in a Polygon Worksheet with examples and problems to find missing angles in various polygons.
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Step-by-step solution for: Angles in Polygons Worksheets - Math Monks
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Show Answer Key & Explanations
Step-by-step solution for: Angles in Polygons Worksheets - Math Monks
To solve for the missing interior angles in each polygon, we need to use the formula for the sum of the interior angles of a polygon, which is:
\[ \text{Sum of interior angles} = (n - 2) \times 180^\circ \]
where \( n \) is the number of sides of the polygon.
This is a quadrilateral (4 sides).
\[ \text{Sum of interior angles} = (4 - 2) \times 180^\circ = 2 \times 180^\circ = 360^\circ \]
Given angles: \( 125^\circ, 68^\circ, 62^\circ \)
Let the missing angle be \( x \).
\[ 125^\circ + 68^\circ + 62^\circ + x = 360^\circ \]
\[ 255^\circ + x = 360^\circ \]
\[ x = 360^\circ - 255^\circ \]
\[ x = 105^\circ \]
This is also a quadrilateral (4 sides).
\[ \text{Sum of interior angles} = (4 - 2) \times 180^\circ = 2 \times 180^\circ = 360^\circ \]
Given angles: \( 105^\circ, 106^\circ, 72^\circ \)
Let the missing angle be \( x \).
\[ 105^\circ + 106^\circ + 72^\circ + x = 360^\circ \]
\[ 283^\circ + x = 360^\circ \]
\[ x = 360^\circ - 283^\circ \]
\[ x = 77^\circ \]
This is a quadrilateral (4 sides).
\[ \text{Sum of interior angles} = (4 - 2) \times 180^\circ = 2 \times 180^\circ = 360^\circ \]
Given angles: \( 98^\circ, 120^\circ, 80^\circ \)
Let the missing angle be \( x \).
\[ 98^\circ + 120^\circ + 80^\circ + x = 360^\circ \]
\[ 298^\circ + x = 360^\circ \]
\[ x = 360^\circ - 298^\circ \]
\[ x = 62^\circ \]
This is a pentagon (5 sides).
\[ \text{Sum of interior angles} = (5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ \]
Given angles: \( x - 27^\circ, x - 15^\circ, x + 30^\circ, 125^\circ, 112^\circ \)
\[ (x - 27^\circ) + (x - 15^\circ) + (x + 30^\circ) + 125^\circ + 112^\circ = 540^\circ \]
\[ 3x + 195^\circ = 540^\circ \]
\[ 3x = 540^\circ - 195^\circ \]
\[ 3x = 345^\circ \]
\[ x = 115^\circ \]
This is a pentagon (5 sides).
\[ \text{Sum of interior angles} = (5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ \]
Given angles: \( x, 115^\circ, 95^\circ, 115^\circ, 130^\circ \)
\[ x + 115^\circ + 95^\circ + 115^\circ + 130^\circ = 540^\circ \]
\[ x + 455^\circ = 540^\circ \]
\[ x = 540^\circ - 455^\circ \]
\[ x = 85^\circ \]
This is a quadrilateral (4 sides).
\[ \text{Sum of interior angles} = (4 - 2) \times 180^\circ = 2 \times 180^\circ = 360^\circ \]
Given angles: \( 60^\circ, x, x, x \)
\[ 60^\circ + x + x + x = 360^\circ \]
\[ 60^\circ + 3x = 360^\circ \]
\[ 3x = 300^\circ \]
\[ x = 100^\circ \]
This is a pentagon (5 sides).
\[ \text{Sum of interior angles} = (5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ \]
Given angles: \( x, 107^\circ, 75^\circ, x - 14^\circ, x + 12^\circ \)
\[ x + 107^\circ + 75^\circ + (x - 14^\circ) + (x + 12^\circ) = 540^\circ \]
\[ 3x + 180^\circ = 540^\circ \]
\[ 3x = 540^\circ - 180^\circ \]
\[ 3x = 360^\circ \]
\[ x = 120^\circ \]
This is a hexagon (6 sides).
\[ \text{Sum of interior angles} = (6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ \]
Given angles: \( 2x - 50^\circ, x + 40^\circ, 150^\circ, 80^\circ, x, x + 20^\circ \)
\[ (2x - 50^\circ) + (x + 40^\circ) + 150^\circ + 80^\circ + x + (x + 20^\circ) = 720^\circ \]
\[ 5x + 240^\circ = 720^\circ \]
\[ 5x = 720^\circ - 240^\circ \]
\[ 5x = 480^\circ \]
\[ x = 96^\circ \]
Final Answer:
1. \( 105^\circ \)
2. \( 77^\circ \)
3. \( 62^\circ \)
4. \( 115^\circ \)
5. \( 85^\circ \)
6. \( 100^\circ \)
7. \( 120^\circ \)
8. \( 96^\circ \)
\[ \text{Sum of interior angles} = (n - 2) \times 180^\circ \]
where \( n \) is the number of sides of the polygon.
Problem 1:
This is a quadrilateral (4 sides).
\[ \text{Sum of interior angles} = (4 - 2) \times 180^\circ = 2 \times 180^\circ = 360^\circ \]
Given angles: \( 125^\circ, 68^\circ, 62^\circ \)
Let the missing angle be \( x \).
\[ 125^\circ + 68^\circ + 62^\circ + x = 360^\circ \]
\[ 255^\circ + x = 360^\circ \]
\[ x = 360^\circ - 255^\circ \]
\[ x = 105^\circ \]
Problem 2:
This is also a quadrilateral (4 sides).
\[ \text{Sum of interior angles} = (4 - 2) \times 180^\circ = 2 \times 180^\circ = 360^\circ \]
Given angles: \( 105^\circ, 106^\circ, 72^\circ \)
Let the missing angle be \( x \).
\[ 105^\circ + 106^\circ + 72^\circ + x = 360^\circ \]
\[ 283^\circ + x = 360^\circ \]
\[ x = 360^\circ - 283^\circ \]
\[ x = 77^\circ \]
Problem 3:
This is a quadrilateral (4 sides).
\[ \text{Sum of interior angles} = (4 - 2) \times 180^\circ = 2 \times 180^\circ = 360^\circ \]
Given angles: \( 98^\circ, 120^\circ, 80^\circ \)
Let the missing angle be \( x \).
\[ 98^\circ + 120^\circ + 80^\circ + x = 360^\circ \]
\[ 298^\circ + x = 360^\circ \]
\[ x = 360^\circ - 298^\circ \]
\[ x = 62^\circ \]
Problem 4:
This is a pentagon (5 sides).
\[ \text{Sum of interior angles} = (5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ \]
Given angles: \( x - 27^\circ, x - 15^\circ, x + 30^\circ, 125^\circ, 112^\circ \)
\[ (x - 27^\circ) + (x - 15^\circ) + (x + 30^\circ) + 125^\circ + 112^\circ = 540^\circ \]
\[ 3x + 195^\circ = 540^\circ \]
\[ 3x = 540^\circ - 195^\circ \]
\[ 3x = 345^\circ \]
\[ x = 115^\circ \]
Problem 5:
This is a pentagon (5 sides).
\[ \text{Sum of interior angles} = (5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ \]
Given angles: \( x, 115^\circ, 95^\circ, 115^\circ, 130^\circ \)
\[ x + 115^\circ + 95^\circ + 115^\circ + 130^\circ = 540^\circ \]
\[ x + 455^\circ = 540^\circ \]
\[ x = 540^\circ - 455^\circ \]
\[ x = 85^\circ \]
Problem 6:
This is a quadrilateral (4 sides).
\[ \text{Sum of interior angles} = (4 - 2) \times 180^\circ = 2 \times 180^\circ = 360^\circ \]
Given angles: \( 60^\circ, x, x, x \)
\[ 60^\circ + x + x + x = 360^\circ \]
\[ 60^\circ + 3x = 360^\circ \]
\[ 3x = 300^\circ \]
\[ x = 100^\circ \]
Problem 7:
This is a pentagon (5 sides).
\[ \text{Sum of interior angles} = (5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ \]
Given angles: \( x, 107^\circ, 75^\circ, x - 14^\circ, x + 12^\circ \)
\[ x + 107^\circ + 75^\circ + (x - 14^\circ) + (x + 12^\circ) = 540^\circ \]
\[ 3x + 180^\circ = 540^\circ \]
\[ 3x = 540^\circ - 180^\circ \]
\[ 3x = 360^\circ \]
\[ x = 120^\circ \]
Problem 8:
This is a hexagon (6 sides).
\[ \text{Sum of interior angles} = (6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ \]
Given angles: \( 2x - 50^\circ, x + 40^\circ, 150^\circ, 80^\circ, x, x + 20^\circ \)
\[ (2x - 50^\circ) + (x + 40^\circ) + 150^\circ + 80^\circ + x + (x + 20^\circ) = 720^\circ \]
\[ 5x + 240^\circ = 720^\circ \]
\[ 5x = 720^\circ - 240^\circ \]
\[ 5x = 480^\circ \]
\[ x = 96^\circ \]
Final Answer:
1. \( 105^\circ \)
2. \( 77^\circ \)
3. \( 62^\circ \)
4. \( 115^\circ \)
5. \( 85^\circ \)
6. \( 100^\circ \)
7. \( 120^\circ \)
8. \( 96^\circ \)
Parent Tip: Review the logic above to help your child master the concept of sum of interior angles of polygons worksheet.
