CHAPTER 11

Areas of Plane Figures

SECTION 11-1

Areas of Rectangles

POSTULATE 17

The area of a square is the square of the length of a side. A = s 2

POSTULATE 18

If two figures are congruent, then they have the same area.

POSTULATE 19

The area of a region is the sum of the areas of its non-overlapping parts.

THEOREM 11-1

The area of a rectangle equals the product of its base and height A = bh

SECTION 11-2

Areas of Parallelograms, Triangles, and Rhombuses

THEOREM 11-2

The area of a parallelogram equals the product of a base and the height to that base.

A = bh

THEOREM 11-3

The area of a triangle equals half the product of a base and the height to that base.

A = ½bh

THEOREM 11-4

The area of a rhombus equals half the product of its diagonals.

A = ½d 1 d 2

SECTION 11-3

Areas of Trapezoids

THEOREM 11-5

The area of a trapezoid equals half the product of the height and the sum of the bases A = ½h(b 1 + b 2 )

Median of a Trapezoid

The segment that joins the midpoints of the legs Median = b 1 + b 2 2

SECTION 11-4

Areas of Regular Polygons

Center of a Regular Polygon

Is the center of the circumscribed circle

Radius of a Regular Polygon

Is the distance from the center to a vertex.

Central Angle of a Regular Polygon

Is an angle formed by two radii drawn to consecutive vertices

Measure of Central Angle of a Regular Polygon

Is 360/n where n is the number of sides

Apothem of a Regular Polygon

Is the perpendicular distance from the center of the polygon to a side

THEOREM 11-6

The area of a regular polygon is equal to half the product of the apothem and the perimeter. A = ½ap

SECTION 11-5

Circumferences and Areas of Circles

Circumference

Is the distance around a circle C = 2  r or  d

Area

A =  r 2

SECTION 11-6

Arc Lengths and Areas of Sectors

Sector of a Circle

Is a region bounded by two radii and an arc of the circle

Length of a Sector

In general, if mAB = x: Length of AB = x/360(2  r)

Area of a Sector

Area of sector AOB = x/360(  r 2 )

SECTION 11-7

Ratios of Areas

Comparing Areas of Triangles 1.

If two triangles have equal heights, then the ratio of their areas equals the ratio of their bases.

Comparing Areas of Triangles 2 . If two triangles have equal bases, then the ratio of their areas equals the ratio of their heights

Comparing Areas of Triangles 3. If two triangles are similar, then the ratio of their areas equals the square of their scale factor.

THEOREM 11-7

If the scale factor of two similar figures is a:b, then 1) The ratio of the perimeters is a:b 2) The ratio of the areas is a 2 :b 2

SECTION 11-8

Geometric Probability

Probability

1.

2.

Suppose a point P of AB is picked at random. Then: Probability that P is on AC = (length of AC)/(length of AB) Suppose a point P of region S is picked at random. Then: Probability that P is in region R = (area of R)/(area of S)

END