Test 8 Review on Area and Polygons

To calculate the area of a parallelogram, you multiply the length of the base by the height. The base is one side of the parallelogram, while the height is the perpendicular distance from this base to the opposite side. This formula captures the total space enclosed within the parallelogram, reflecting how wide and tall it is, which is essential for accurately determining its area.

The area of a trapezoid is calculated by averaging the lengths of the two parallel sides (bases) and multiplying by the height. The formula A = 1/2 * (base1 + base2) * height effectively captures this relationship, as it first sums the lengths of the bases, divides by two to find the average base length, and then multiplies by the height to account for the vertical dimension, resulting in the total area of the trapezoid.

The area of a circle is calculated using the formula A = π * radius^2, where π (pi) is a constant approximately equal to 3.14. This formula arises from the relationship between the radius and the total space enclosed within the circle. By squaring the radius and multiplying by π, we account for the circular shape, allowing us to determine the total area effectively. Other formulas listed pertain to different geometric shapes, such as rectangles and triangles, and are not applicable to circles.

The area of a triangle is calculated using the formula A = 1/2 * base * height because it takes into account the two essential dimensions of the triangle: its base and its height. The height is the perpendicular distance from the base to the opposite vertex, and by multiplying the base by the height, you find the area of a rectangle that would encompass the triangle. Dividing by 2 adjusts for the fact that a triangle occupies half the area of that rectangle, thus providing the correct measurement for the triangle's area.

The area of a regular polygon can be calculated using the formula A = (1/2) * Perimeter * Apothem. This formula works by taking half of the perimeter, which represents the total length of all the sides, and multiplying it by the apothem, the distance from the center to the midpoint of a side. This approach effectively divides the polygon into triangles, allowing for an accurate calculation of the total area by summing the areas of these triangles.

The total interior angle sum of a polygon can be derived from the fact that a polygon can be divided into triangles. Since each triangle has an angle sum of 180°, and a polygon with n sides can be divided into (n - 2) triangles, the total sum of the interior angles is (n - 2) multiplied by 180°. This formula applies to any polygon, regardless of the number of sides, providing a consistent method to calculate the interior angle sum.

In a regular hexagon, all sides and angles are equal. To find the measure of each interior angle, we use the formula: \((n-2) \times 180° / n\), where \(n\) is the number of sides. For a hexagon, \(n = 6\). Plugging in the values: \((6-2) \times 180° / 6 = 4 \times 180° / 6 = 720° / 6 = 120°\). Thus, each interior angle in a regular hexagon measures 120°.

The formula for the length of an arc of a circle is derived from the relationship between the circle's total circumference and the angle that subtends the arc. The full circumference is given by \(2 \pi \times \text{radius}\). To find the length of an arc corresponding to a specific angle \(θ\) in degrees, we take the fraction of the circle represented by that angle, which is \(θ/360\), and multiply it by the total circumference, resulting in \(L = (θ/360) \times 2 \pi \times \text{radius}\).

The area of a sector of a circle is determined by the proportion of the circle represented by the angle θ. Since a full circle corresponds to 360 degrees, the formula A = (θ/360) * π * radius^2 calculates the area based on the fraction of the circle's total area that the sector occupies. Here, π * radius^2 gives the area of the entire circle, and multiplying by (θ/360) adjusts it to reflect just the sector's size relative to the whole circle.

To find the perimeter of a regular polygon, you multiply the length of one side by the total number of sides. In this case, the polygon has 8 sides, and each side measures 5 cm. Therefore, the perimeter is calculated as 8 sides × 5 cm/side = 40 cm. This straightforward multiplication gives the total distance around the polygon, confirming that the perimeter is indeed 40 cm.