# polygon and polygon shape

A polygon shape is a closed two-dimensional shape with straight sides, where each side meets another at a vertex. polygon shape It encompasses diverse sorts, which consist of triangles, quadrilaterals, and hexagons. polygon shape Polygons are characterized by the extensive variety of factors and angles they own, making them an important idea in geometry.

**Problem 1: Triangle Area**

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**Problem: Find the area of a triangle with a base of 8 inches and a height of 6 inches.**

Solution: The area (A) of a triangle is calculated as (1/2) * base * height. Substituting the values: A = (1/2) * 8 inches * 6 inches = 24 square inches.

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**Problem 2: Perimeter of a Square**

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**Problem: Determine the perimeter of a square with a side length of 10 centimeters.**

Solution: The perimeter (P) of a square is found by multiplying the side length by 4. So, P = 10 cm * 4 = 40 centimeters.

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**Problem 3: Sum of Interior Angles**

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**Problem: Calculate the sum of interior angles in a hexagon.**

Solution: To find the sum of interior angles in a polygon, use the formula: (n-2) * 180 degrees, where “n” is the number of sides. For a hexagon: (6-2) * 180 degrees = 4 * 180 degrees = 720 degrees.

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**Problem 4: Area of a Parallelogram**

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**Problem: Find the area of a parallelogram with a base of 12 meters and a height of 5 meters.**

Solution: The area (A) of a parallelogram is given by the formula base * height. Substituting the values: A = 12 meters * 5 meters = 60 square meters.

**Problem 5: Diagonals of a Rectangle**

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**Problem: Calculate the length of the diagonals of a rectangle with sides of 8 inches and 6 inches.**

Solution: The length of the diagonals in a rectangle can be found using the Pythagorean theorem. Diagonal = âˆš(lengthÂ² + widthÂ²) = âˆš(8 inchesÂ² + 6 inchesÂ²) â‰ˆ 10 inches.

## polygon and polygon shape

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**Problem 6: Exterior Angle of a Pentagon**

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**Problem: Determine the measure of the exterior angle of a regular pentagon.**

Solution: In a regular polygon, all exterior angles have the same measure. To find it, use the formula: 360 degrees/number of sides. For a pentagon: 360 degrees / 5 = 72 degrees.

**Problem 7: Perimeter of a Triangle**

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**Problem: Calculate the perimeter of a triangle with sides of lengths 7 cm, 9 cm, and 12 cm.**

Solution: Add the lengths of all three sides to find the perimeter. P = 7 cm + 9 cm + 12 cm = 28 cm.

**Problem 8: Area of a Rhombus**

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**Problem: Find the area of a rhombus with diagonals of lengths 10 meters and 12 meters.**

Solution: The area (A) of a rhombus is given by the formula: (1/2) * product of diagonals. Substituting the values: A = (1/2) * 10 meters * 12 meters = 60 square meters.

**Problem 9: Interior Angle of a Regular Decagon**

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**Problem: Determine the measure of an interior angle in a regular decagon.**

Solution: To measure an interior angle in a regular polygon, use the formula: 180 degrees * (n-2) / n, where “n” is the number of sides. For a decagon: 180 degrees * (10-2) / 10 = 144 degrees.

**Problem 10: Area of a Trapezoid**

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**Problem: Find the area of a trapezoid with bases of lengths 5 cm and 9 cm and a height of 8 cm.**

Solution: The area (A) of a trapezoid is calculated as (1/2) * (sum of bases) * height. Substituting the values: A = (1/2) * (5 cm + 9 cm) * 8 cm = 56 square centimeters.

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**Question 11: Circumference of a Circle**

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**Question: Calculate the circumference of a circle with a radius of 6 centimeters.**

Solution: The circumference (C) of a circle is given by the formula C = 2Ï€r, where “r” is the radius. Substituting the value: C = 2Ï€ * 6 cm â‰ˆ 37.70 cm.

**Question 12: Area of a Circle**

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**Question: Find the area of a circle with a diameter of 10 meters.**

Solution: The area (A) of a circle is calculated as A = Ï€rÂ², where “r” is the radius (half of the diameter). Substituting the value: A = Ï€ * (10/2 meters)Â² = 25Ï€ square meters â‰ˆ 78.54 square meters.

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**Question 13: Perimeter of a Regular Hexagon**

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**Question: Determine the perimeter of a regular hexagon with a side length of 8 inches.**

Solution: In a regular hexagon, all sides have the same length. So, the perimeter (P) is found by multiplying the side length by 6 (the number of sides). P = 8 inches * 6 = 48 inches.

## polygon and polygon shape

**Question 14: Area of a Sector**

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**Question: Find the area of a sector with a central angle of 45 degrees in a circle with a radius of 12 centimeters.**

Solution: The area (A) of a sector is calculated as (Î¸/360) * Ï€rÂ², where “Î¸” is the central angle in degrees, and “r” is the radius. Substituting the values: A = (45/360) * Ï€ * (12 cm)Â² â‰ˆ 56.55 square centimeters.

**Question 15: Volume of a Cube**

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**Question: Calculate the volume of a cube with an edge length of 5 centimeters.**

Solution: The volume (V) of a cube is given by V = aÂ³, where “a” is the edge length. Substituting the value: V = (5 cm)Â³ = 125 cubic centimeters.

**Question 16: Surface Area of a Cylinder**

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**Question: Find the total surface area of a cylinder with a radius of 4 meters and a height of 10 meters.**

Solution: The total surface area (SA) of a cylinder is calculated as SA = 2Ï€rÂ² + 2Ï€rh, where “r” is the radius, and “h” is the height. Substituting the values: SA = 2Ï€ * (4 m)Â² + 2Ï€ * 4 m * 10 m â‰ˆ 251.2 square meters.

**Question 17: Volume of a Sphere**

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**Question: Determine the volume of a sphere with a radius of 7 centimeters.**

Solution: The volume (V) of a sphere is given by V = (4/3)Ï€rÂ³, where “r” is the radius. Substituting the value: V = (4/3)Ï€ * (7 cm)Â³ â‰ˆ 1436.76 cubic centimeters.

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**Question 18: Similar Triangles**

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**Question: Two triangles are similar. If one triangle has a base of 8 inches and a height of 12 inches, and the other has a base of 10 inches, find the height of the second triangle.**

Solution: Since the triangles are similar, the ratio of corresponding sides is the same. So, (8 inches / 12 inches) = (10 inches / x), where “x” is the height of the second triangle. Solve for “x”: x = (10 inches * 12 inches) / 8 inches = 15 inches.

**Question 19: Area of a Trapezoid**

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**Question: Calculate the area of a trapezoid with bases of lengths 6 meters and 8 meters and a height of 5 meters.**

Solution: The area (A) of a trapezoid is given by A = (1/2) * (sum of bases) * height. Substituting the values: A = (1/2) * (6 m + 8 m) * 5 m = 35 square meters.

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**Question 20: Exterior Angle Sum of a Polygon**

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**Question: Find the sum of the exterior angles in a polygon with 10 sides.**

Solution: The sum of the exterior angles in any polygon, regardless of the number of sides, is always 360 degrees. So, for a polygon with 10 sides, the sum is also 360 degrees.

**Question 21: Diagonals of a Pentagon**

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**Question: Calculate the number of diagonals in a regular pentagon.**

Solution: The number of diagonals in a polygon with “n” sides is given by the formula (n(n-3))/2. For a pentagon: (5(5-3))/2 = 5 diagonals.

**Question 22: Area of a Quadrilateral**

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**Question: Find the area of a quadrilateral with side lengths of 7 cm, 8 cm, 9 cm, and 10 cm.**

Solution: Divide the quadrilateral into two triangles. Calculate the area of each triangle separately, and then sum them up. Using Heron’s formula for each triangle, the total area is approximately 34.57 square centimeters.

**Question 23: Interior Angle Sum of an Irregular Polygon**

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**Question: Calculate the sum of the interior angles in an irregular polygon with 9 sides.**

Solution: To find the sum of interior angles in an irregular polygon, use the formula: (n-2) * 180 degrees, where “n” is the number of sides. For a polygon with 9 sides: (9-2) * 180 degrees = 1260 degrees.

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**Question 24: Pythagorean Theorem**

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**Question: In a right triangle, if one leg is 3 units long and the hypotenuse is 5 units long, what is the length of the other leg?**

Solution: Use the Pythagorean theorem: aÂ² + bÂ² = cÂ², where “a” and “b” are the lengths of the legs, and “c” is the length of the hypotenuse. Substituting the values: 3Â² + bÂ² = 5Â². Solve for “b”: bÂ² = 25 – 9 = 16, so b = 4 units.

**Question 25: Volume of a Cone**

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**Question: Determine the volume of a cone with a radius of 9 inches and a height of 12 inches.**

Solution: The volume (V) of a cone is given by V = (1/3)Ï€rÂ²h, where “r” is the radius and “h” is the height. Substituting the values: V = (1/3)Ï€ * (9 inches)Â² * 12 inches â‰ˆ 3053.63 cubic inches.

**Question 26: Surface Area of a Rectangular Prism**

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**Question: Find the total surface area of a rectangular prism with dimensions of 6 meters, 8 meters, and 10 meters.**

Solution: The total surface area (SA) of a rectangular prism is given by SA = 2lw + 2lh + 2wh, where “l,” “w,” and “h” are the length, width, and height, respectively. Substituting the values: SA = 2(6 m * 8 m) + 2(6 m * 10 m) + 2(8 m * 10 m) = 376 square meters.

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**Question 27: Ratio of Areas**

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**Question: Two similar triangles have areas in the ratio of 9:16. If the area of the smaller triangle is 36 square inches, find the area of the larger triangle.**

Solution: Let “x” represent the area of the larger triangle. Using the given ratio, 36 square inches (smaller triangle) / x (larger triangle) = 9/16. Solve for “x”: x = (36 square inches * 16) / 9 = 64 square inches.

**Question 28: Angle Bisector Theorem**

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**Question: In triangle ABC, the angle bisector of angle A divides side BC into segments of lengths 5 cm and 3 cm. Find the length of side AC if side AB is 10 cm.**

Solution: According to the Angle Bisector Theorem, the ratio of the lengths of the two segments of side BC is equal to the ratio of the lengths of the two sides emanating from angle A. So, (BC/5 cm) = (AB/AC). Substituting the values: (8 cm/5 cm) = (10 cm/AC). Cross-multiply and solve for AC: AC = (5 cm * 10 cm) / 8 cm = 6.25 cm.

**Question 29: Radius of Inscribed Circle**

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**Question: Given a triangle with sides of lengths 7 cm, 24 cm, and 25 cm, find the radius of the inscribed circle.**

Solution: The radius (r) of the inscribed circle can be found using the formula r = (Area of the triangle) / (Semiperimeter of the triangle). First, calculate the semiperimeter: s = (7 cm + 24 cm + 25 cm) / 2 = 28 cm. Then, find the area using Heron’s formula: A = âˆš[s(s-a)(s-b)(s-c)]. Finally, use r = A / s to find the radius.

**Question 30: Exterior Angle of a Quadrilateral**

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**Question: Calculate the measure of an exterior angle of a quadrilateral.**

Solution: The measure of an exterior angle in any polygon is always equal to 360 degrees divided by the number of sides. For a quadrilateral (4 sides), each exterior angle measures 360 degrees / 4 = 90 degrees.