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.

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.

 

 

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

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.

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.

Solution: The length of the diagonals in a rectangle can be found using the Pythagorean theorem. Diagonal = √(length² + width²) = √(8 inches² + 6 inches²) ≈ 10 inches.

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.

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

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.

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.

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.

Solution: The circumference (C) of a circle is given by the formula C = 2πr, where “r” is the radius. Substituting the value: C = 2π * 6 cm ≈ 37.70 cm.

Solution: The area (A) of a circle is calculated as A = πr², where “r” is the radius (half of the diameter). Substituting the value: A = π * (10/2 meters)² = 25π square meters ≈ 78.54 square meters.

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.

Solution: The area (A) of a sector is calculated as (θ/360) * πr², where “θ” is the central angle in degrees, and “r” is the radius. Substituting the values: A = (45/360) * π * (12 cm)² ≈ 56.55 square centimeters.

Solution: The volume (V) of a cube is given by V = a³, where “a” is the edge length. Substituting the value: V = (5 cm)³ = 125 cubic centimeters.

Solution: The total surface area (SA) of a cylinder is calculated as SA = 2πr² + 2πrh, where “r” is the radius, and “h” is the height. Substituting the values: SA = 2π * (4 m)² + 2π * 4 m * 10 m ≈ 251.2 square meters.

Solution: The volume (V) of a sphere is given by V = (4/3)πr³, where “r” is the radius. Substituting the value: V = (4/3)π * (7 cm)³ ≈ 1436.76 cubic centimeters.

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.

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.

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.

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.

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.

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.

Solution: Use the Pythagorean theorem: a² + b² = c², where “a” and “b” are the lengths of the legs, and “c” is the length of the hypotenuse. Substituting the values: 3² + b² = 5². Solve for “b”: b² = 25 – 9 = 16, so b = 4 units.

Solution: The volume (V) of a cone is given by V = (1/3)πr²h, where “r” is the radius and “h” is the height. Substituting the values: V = (1/3)π * (9 inches)² * 12 inches ≈ 3053.63 cubic inches.

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.

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.

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.

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 = √[s(s-a)(s-b)(s-c)]. Finally, use r = A / s to find the radius.

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.