In this section, we will focus on finding the area of a parallelogram and also finding missing dimensions given the area.
Area of a Parallelogram
The formula to find the area of a parallelogram is A = bh, where A represents the area and b and h represent the base and height of the parallelogram respectively. A polygon is a closed figure formed by three or more straight lines, while a parallelogram is a quadrilateral with opposite sides parallel and of the same length. Additionally, a rhombus is a parallelogram with four equal sides.
Understanding the Base and Height
The base of a parallelogram can be any one of its sides, which can either be the top or bottom side. The height is the perpendicular distance from the base to the opposite side.
Examples
To understand this better, let's take a look at a few examples:
Find the area of the parallelogram where the base (b) is 5cm and the height (h) is 8cm. Using the formula A = bh, we can calculate the area as A = 5 * 8 = 40cm².
Find the base of a parallelogram given that the area is 17cm², and the height is 8cm. Using the formula A = bh, we can rearrange it to find the base, which is 17/8 = 2.125cm.
Austin is building a dog run, and the space he will enclose is in the shape of a parallelogram. To find the area of the space, given the base (b) as 10ft and the height (h) as 6.25ft, we can use the formula A = bh to calculate the area as 10 * 6.25 = 65.625ft².
Area of Triangles
Moving on to triangles, in this section we will learn to find the area of a triangle and missing dimensions given the area.
Area of a Triangle
The formula to find the area of a triangle is A = 1/2 * bh, where A represents the area, b is the base of the triangle, and h is the height of the triangle.
Examples
Let's take a look at a few examples to understand how to find the area of triangles:
Find the area of a triangle where the base is 16cm and the height is 8cm. Using the formula A = 1/2 * bh, we can calculate the area as A = 1/2 * 16 * 8 = 64cm².
Find the height of a triangle where the area is 40m² and the base is 8m. Rearranging the formula A = 1/2 * bh, we can calculate the height as 40 = 1/2 * 8 * h, which gives us h = 10m.
Area of Trapezoids
In this section, we will focus on finding the area of a trapezoid and missing dimensions given the area.
Area of a Trapezoid
The formula to find the area of a trapezoid is A = 1/2 * h * (b₁ + b₂), where A represents the area, h is the height of the trapezoid, and b₁ and b₂ are the lengths of the parallel sides.
Examples
Let's take a look at an example to understand how to find the area of a trapezoid:
Find the area of a trapezoid where the height is 3m, and the lengths of the parallel sides are 4m and 7.6m. Using the formula A = 1/2 * h * (b₁ + b₂), we can calculate the area as A = 1/2 * 3 * (4 + 7.6) = 17.4m².
Changes in Dimension
In this section, we will learn about the effects dimensions have on the perimeter and area of polygons.
Effects on Perimeter
The perimeter of a polygon changes by a factor when the dimensions of the polygon are multiplied by x. For example, if the dimensions of Figure A are multiplied by 2 to produce Figure B, the perimeter of Figure B will be 2 times the perimeter of Figure A.
Effects on Area
When the dimensions of a polygon are multiplied by x, the area of the polygon changes by x². For example, if the area of Figure A is 20 and its dimensions are multiplied by 2, the area of Figure B would be 80, which is 2² times the area of Figure A.
Polygons on the Coordinate Plane
In this section, we will learn to find the perimeter and area of polygons on a coordinate plane.
Examples
A rectangle has vertices at coordinates (1, 3), (1, 7), (3, 7), and (3, 3). The length of each side can be found using the distance formula. In this case, the perimeter is 12 units, and the area is 8 square units.
Finding the area of a figure involves separating the figure into different shapes, finding the area of each individual shape, and then adding the areas together. This method is applicable in finding the area of composite figures and overlapping figures.
Area of Composite Figures
Composite figures are made up of two or more two-dimensional shapes. Finding the area of such figures requires adding the areas of the individual shapes.
Example
The diagram shows a composite figure made up of a rectangle and a trapezoid. By finding the areas of the individual shapes and adding them together, the total area of the composite figure can be calculated.
By understanding these concepts and practicing with various examples, one can gain a strong understanding of how to calculate the area and perimeter of different shapes, as well as the effects of dimension changes on these calculations.
