Chapter 10

MeasurementSection 10.3

Areas of Quadrilaterals, Triangles and Circles

Areas of Other Shapes

The areas of other (often more complicated) shapes can be found by either breaking them apart to form familiar (less complicated) shapes or by copying them and assembling them into a familiar (less complicated) shape.

Parallelogram

Finding the area of a parallelogram can be done by cutting it apart into a triangle and a trapezoid then translating the triangle part into another position to form a rectangle.

Both of these examples illustrate that for a parallelogram the area is the same as a rectangle where you need to measure the length of one of the sides called the base (b) and the perpendicular distance between the sides called the height (h)

h

b

Area = (base)·(height) = bh

The area of the parallelogram pictured on this geoboard can be found in the following way:

Area = 4·2 = 8 square units

In this case 4 is the base and 2 is the height.

4

2

Triangles

The area of a triangle can be thought of as half the area of a parallelogram or two congruent triangles make a parallelogram.

These three examples show the area of a triangle is half the area of the parallelogram formed by 2 congruent triangles. In this case the height (h) is the perpendicular distance from one vertex to the other side which is the base (b). (Note: The height does not always need to be inside the triangle.)

b

h

Area = bh2

1

Find the areas of the green and blue triangles.

The area of the green triangle is:

Area = ½ · (4·3) = ½ · 12 = 6 square units

The area of the blue triangle is:

Area = ½ · (3·3) = ½ · 9 = 4.5 square units

3

3

34

Trapezoids

The area of a trapezoid can be thought of as half of a parallelogram made out of two congruent trapezoids.

The area of a trapezoid is half the area of the two parallelograms. The trapezoid will have 2 bases (b1 and b2) formed by the two parallel sides and one height (h) which is the perpendicular distance between the two parallel sides.

b1

b2

h

Area = ½ · h (b1 + b2)

Circles

The area of a circle is more difficult to illustrate. The idea is to cut it into pieces that are called sectors (they look like pie slices) and form something that is “close” to being a parallelogram.

r

½ C = ½ d = ½ 2r = r

Area r2

Cut up even smaller:

Area = r2 square units

Just like for perimeter the area of a circle can be divided up proportionately. Half the circle will have half the area. A quarter circle will have a quarter the area. Use this to find the area of the shape to the right.

Divide up the shape into a square and 2 shapes that are ¾ of a circle.

Area = 2·2+ ¾ ··22+ ¾ ··22 = 4 + ¾ ·4+ ¾ ·4

= 4 + 3 + 3 = 4 + 6 square units

One of the methods that can be used to find the area of a shape on a geoboard is to enclose the shape with something you know the area of. Subtract off the parts that are not part of the original shape.

Use this to find the area of the triangle to the left.

1. Enclose the triangle with a green rectangle.

2. Find the area of the triangle in the upper left corner.

Area = (½)·2·2 = 2

3. Find the area of the triangle in the upper right corner.

Area = (½)·3·3 = 92 = 4.5

4. Find the area of the triangle in the lower left corner

Area = (½)·1·5 = 52 = 2.5

5. Subtract the areas from steps (2), (3) and (4) from the area of the rectangle to get the area of the triangle.

Area = (Area of rectangle) – Area of 3 triangles = 15 – (2 + 4.5 + 2.5) = 15 - 9 = 6

2

2 3

3

5

1