Area of Polygons

1. Area of Polygons By Sara Gregurash

2. Area • The area of a polygon measures the size of the region that the figure occupies. It is 2-dimensional like a table cloth. It is expressed in terms of some square unit. Some examples of the units used are square meters, square centimeters, square inches, or square kilometers.

3. Area of a triangle • To find the area of a triangle, you have to multiply the base by its height and divide by 2. You have to divide by two because a parallelogram can be divided into 2 triangles. The area of each triangle (2 of them) of a parallelogram is equal to one-half the area of the parallelogram. The equation is: A = ½ (bh) • The b stands for the base of a triangle. The h stands for height. These are shown in the two diagrams below.

4. Base and Height • The base and height of a triangle have to be perpendicular. The base is a side of the triangle, but the height isn’t always a side of the triangle. When you have a right triangle, then the height is the side of the triangle because it is perpendicular to the base. The sides of a triangle aren’t always perpendicular to the base. For example, when you have an obtuse triangle the side is slanted (this is shown below). h = 8 cm b = 4 cm

5. Area of a right triangle • Let’s practice finding the area of triangles. The same formula is used to find the area of all triangles. I will show you how to find the area of right, acute, obtuse, scalene, and equilateral triangles. Here is an example using right triangles. • For this triangle, the base is 6 cm and the height is 9 cm so you plug those numbers into the equation. It looks like this: A = ½ (2 in · 4 in) A = ½ (8 in2) A = 4 in2 h = 4 in b = 2 in

6. Area of an acute and obtuse triangle • The base of this acute triangle is 16 centimeters and the height is 7 centimeters. A = ½ (16 cm · 7 cm) A = ½ (112 cm2) A = 56 cm2 • The base of this obtuse triangle is 4 inches and the height is 8 inches. A = ½ (8 in · 4 in) A = ½ (32 in2) A = 16 in2 b = 7 cm b = 16 cm h = 8 cm b = 4 cm

7. Area of a scalene triangle • A scalene triangle has all sides of different lengths. • The base of this scalene triangle is 80 millimeters and the height is 22 millimeters. A = ½ (b · h) A = ½ (80 mm · 22 mm) A = ½ (1760 mm 2 ) A = 880 mm 2 b = 80 mm h = 22 mm s =50 mm

8. Area of an equilateral triangle • An equilateral triangle is a triangle with three equal length sides and the angles are all equal. • To find the area of the equilateral triangle below with a side of 21 centimeters and the altitude is 14 centimeters, you have to use the formula. A = ½ base or side · altitude (height) A = ½ (21 cm · 14 cm) A = ½ (294 cm 2 ) A = 147 cm 2 side (base) = 21 cm a = 14 cm Math Goodies Lesson is a good site to view to practice your skills on area of triangles.

9. Area of a square • A square is a rectangle with 4 equal sides. To find the area of a square you multiply the distance along the side of the square by itself. The formula looks like this:A = s · s (s is the distance along the side of the square) • So, the length of each side of this square is 2 inches. A = 2 in · 2 in A = 4 in2 2 in

10. Area of a rectangle • A rectangle is a four-sided polygon that has opposite sides equal and are parallel. All four angles are right angles that are 90 degrees. • To find the area of a rectangle you have to multiply the length by the width. The formula looks like this: A = L · W (L is the length and W is the width of the rectangle) • So, the length of this rectangle is 19 centimeters and the width is 6 centimeters. A = 19 cm · 6 cm A = 114 cm2 6 cm 19 cm This sites has a lesson on area of squares and rectangles.

11. Parallelograms • A parallelogram is a four-sided shape formed by two pairs of parallel lines. The opposite sides have the same length and the opposite angles have the same degree (size). • To find the area of a parallelogram, you have to multiply its base by its height. The formula looks like this: A =b · h (b is the base and h is the height) • The base and height of a parallelogram must be perpendicular. The lateral sides of a parallelogram are not perpendicular to the base, so a dotted line is drawn to represent the height. h b

12. Area of a parallelogram • To find the area of this parallelogram, you multiply the base of 8 centimeters and the height of 3 centimeters. A = b · h A = 8 cm · 3 cm A = 24 cm2 h = 3 cm b = 8 cm This site has a lesson about the area of parallelograms.

13. Trapezoids • A trapezoid is a four-sided figure with one pair of parallel sides. The bases of the trapezoid below are parallel. To find the area of a trapezoid, you have to multiply the sum of its bases by the height and divided by 2. The formula looks like this: A = 1/2 · (b1 + b2) · h ( b1 is base1, b2 is base2, and h is the height) • Each base of a trapezoid must be perpendicular to the height. The lateral sides of the trapezoid aren’t perpendicular to either of the bases, so a dotted line is drawn to represent the height. b1 h b2

14. Area of a trapezoid • The trapezoid below has a b1 of14 millimeters, b2 of 20 millimeters, and a height of 22 millimeters. So, the area of this trapezoid is: A = 1/2 · (b1 + b2) · h A = 1/2 · (14 mm + 20 mm) · 22 mm A = 1/2 (34 mm) 22 mm A = 1/2 · 748 mm2 A = 374 mm2 b1= 14 mm h = 22 mm b2 = 20 mm This site has a lesson on the area of trapezoids.

15. Practice Websites • These sites have great examples of how to find the area of polygons and circles. The second one is a worksheet to help you practice your skills. It is a good idea to take a look at them and see if you learn more about area. Then you can take the quiz that it provides. After you do that, you can move on to my worksheet for more practice. Area of Polygons and Circles Worksheet on Area of Polygons and Circles

16. Worksheet Time • Now, that you have went through my lesson about areas of polygons. So, you can apply what you learned with this worksheet. Time to practice! • Here are the answers to my worksheet.