A tangram is an ancient Chinese puzzle made from a square. The pieces can be rearranged to form many different shapes. The area of a figure made with all the pieces is the sum of the areas of the pieces.

Remember that rectangles and squares are also parallelograms. The area of a square with side s is A = s 2, and the perimeter is P = 4s.

The height of a parallelogram is measured along a segment perpendicular to a line containing the base. Remember!

The perimeter of a rectangle with base b and height h is P = 2b + 2h or P = 2 (b + h). Remember!

The diagonals of a rhombus or kite are perpendicular, and the diagonals of a rhombus bisect each other. Remember!

A circle is the locus of points in a plane that are a fixed distance from a point called the center of the circle. A circle is named by the symbol  and its center.  A has radius r = AB and diameter d = CD. Solving for C gives the formula C = d. Also d = 2r, so C = 2r. The irrational number  is defined as the ratio of the circumference C to the diameter d, or

The area of a sector is a fraction of the circle containing the sector. To find the area of a sector whose central angle measures m°, multiply the area of the circle by

Find the area of each sector. Give answers in terms of  and rounded to the nearest hundredth. Example 1B: Finding the Area of a Sector sector ABC Use formula for area of sector. Substitute 5 for r and 25 for m.  1.74 ft 2  5.45 ft 2 Simplify.

In the same way that the area of a sector is a fraction of the area of the circle, the length of an arc is a fraction of the circumference of the circle.

Find each arc length. Give answers in terms of  and rounded to the nearest hundredth. Example 4A: Finding Arc Length FG Use formula for area of sector.  5.96 cm  cm Substitute 8 for r and 134 for m. Simplify.

Lesson Quiz: Part I Find each measure. Give answers in terms of  and rounded to the nearest hundredth. 1. area of sector LQM 2.5  in.  7.85 in. 7.5 in 2  in 2 2. length of NP

The center of a regular polygon is equidistant from the vertices. The apothem is the distance from the center to a side. A central angle of a regular polygon has its vertex at the center, and its sides pass through consecutive vertices. Each central angle measure of a regular n-gon is

Regular pentagon DEFGH has a center C, apothem BC, and central angle DCE.

To find the area of a regular n-gon with side length s and apothem a, divide it into n congruent isosceles triangles. The perimeter is P = ns. area of each triangle: total area of the polygon:

The tangent of an angle in a right triangle is the ratio of the opposite leg length to the adjacent leg length. See page 525. Remember!

A composite figure is made up of simple shapes, such as triangles, rectangles, trapezoids, and circles. To find the area of a composite figure, find the areas of the simple shapes and then use the Area Addition Postulate.

Find the shaded area. Round to the nearest tenth, if necessary. Example 1A: Finding the Areas of Composite Figures by Adding Divide the figure into parts. area of half circle:

Example 1A Continued area of the rectangle: area of triangle: shaded area: A = bh = 20(14) = 280 mm 2 50 ≈ mm 2

Example 2: Finding the Areas of Composite Figures by Subtracting Find the shaded area. Round to the nearest tenth, if necessary. area of circle: A = r 2 = (10) 2 = 100 cm 2 area of trapezoid: area of figure: 100 –128  cm 2

Remember!

Describe the effect of each change on the area of the given figure. Example 1: Effects of Changing One Dimension The height of the triangle is multiplied by 6. original dimensions: multiply the height by 6: Notice that 180 = 6(30). If the height is multiplied by 6, the area is also multiplied by 6. = 30 in 2 = 180 in 2

If the radius of a circle or the side length of a square is changed, the size of the entire figure changes proportionally. Helpful Hint

When the dimensions of a figure are changed proportionally, the figure will be similar to the original figure.

Remember that in probability, the set of all possible outcomes of an experiment is called the sample space. Any set of outcomes is called an event. If every outcome in the sample space is equally likely, the theoretical probability of an event is

Geometric probability is used when an experiment has an infinite number of outcomes. In geometric probability, the probability of an event is based on a ratio of geometric measures such as length or area. The outcomes of an experiment may be points on a segment or in a plane figure.

If an event has a probability p of occurring, the probability of the event not occurring is 1 – p. Remember!

Use the spinner to find the probability of each event. Example 3A: Using Angle Measures to Find Geometric Probability the pointer landing on yellow The angle measure in the yellow region is 140°.