Name Date Class. Developing Formulas for Triangles and Quadrilaterals. Triangle. A bh

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1 Chapter 9. Developing Formulas for Triangles and Quadrilaterals Area of Triangles and Quadrilaterals Parallelogram Triangle Trapezoid A bh A bh A b b h Find the perimeter of the rectangle in which A 7 mm. Step Find the height. A bh Area of a rectangle 7 3h Substitute 7 for A and 3 for b. 9 mm h Divide both sides by 3. Step Use the base and the height to find the perimeter. P b h Perimeter of a rectangle P (3) (9) 4 mm Substitute 3 for b and 9 for h. Find each measurement.. the area of the parallelogram. the base of the rectangle in which A 36 mm 3. the area of the trapezoid 4. the height of the triangle in which A 9 cm 5. the perimeter of a rectangle in which 6. b of a trapezoid in which A 5 ft, A 54 in and h in. h ft, and b ft

2 Chapter 9. Developing Formulas for Triangles and Quadrilaterals continued Area of Rhombuses and Kites Rhombus Kite A dd A dd Find d of the kite in which A 56 in. A dd Area of a kite 56 6 d Substitute 56 in for A and 6 in. for d. 56 3d Simplify. in. d Divide both sides by 3. Find each measurement. 7. the area of the rhombus 8. d of the kite in which A 44 ft 9. d of the rhombus in which A 90 m 0. d of the kite in which A 39 mm. d of a kite in which A 6x m and. the area of a rhombus in which d 8 m d 4ab in. and d 7a in.

3 Chapter 9. Developing Formulas for Circles and Regular Polygons Circumference and Area of Circles A circle with diameter d and radius r has circumference C d or C r. A circle with radius r has area A r. Find the circumference of circle S in which A 8 cm. Step Use the given area to solve for r. Step A r Area of a circle 8 cm r Substitute 8 for A. 8 cm r Divide both sides by. 9 cm r Take the square root of both sides. Use the value of r to find the circumference. C r C (9 cm) 8 cm Circumference of a circle Substitute 9 cm for r and simplify. Find each measurement.. the circumference of circle B. the area of circle R in terms of 3. the area of circle Z in terms of 4. the circumference of circle T in terms of 5. the circumference of circle X in 6. the radius of circle Y in which C 8 cm which A 49 in

4 Chapter 9. Developing Formulas for Circles and Regular Polygons continued Area of Regular Polygons The area of a regular polygon with apothem a and perimeter P is A ap. The center is equidistant from the vertices. The apothem is the distance from the center to a side. Find the area of a regular hexagon with side length 0 cm. Step Draw a figure and find the measure of a central angle. Each central 360 angle measure of a regular n-gon is n. A central angle has its vertex at the center. This central angle measure is n Step Step 3 Use the tangent ratio to find the apothem. You could also use the Thm. in this case. leg opposite 30 angle tan 30 Write a tangent ratio. leg adjacent to 30 angle tan 30 5 cm a Substitute the known values. 5 cm a tan 30 Solve for a. Use the formula to find the area. A ap 5 5 A 60 a, P 6 0 or 60 cm tan 30 tan30 A 59.8 cm Simplify. Find the area of each regular polygon. Round to the nearest tenth a regular hexagon with an apothem of 3 m 0. a regular decagon with a perimeter of 70 ft

5 Chapter 9.3 Composite Figures The figure at right is called a composite figure because it is made up of simple shapes. To find its area, first find the areas of the simple shapes and then add. Divide the figure into a triangle and a rectangle. The base of the triangle is cm. The height of the triangle is 7 4 cm. area of triangle: A bh area of rectangle: A bh 5 4 8(7) 0 cm 6 cm The area of the whole figure is cm. Find the shaded area. Round to the nearest tenth if necessary

6 Chapter 9.3 Composite Figures continued You can also find the area of composite figures by using subtraction. To find the area of the figure at right, subtract the area of the square from the area of the rectangle. area of rectangle: A bh A s (9) 4 area of square: 08 in 6 in The shaded area is in. Find the shaded area. Round to the nearest tenth if necessary

7 Chapter 9.4 Perimeter and Area in the Coordinate Plane One way to estimate the area of irregular shapes in the coordinate plane is to count the squares on the grid. You can estimate the number of whole squares and the number of half squares and then add. The polygon with vertices A( 3, ), B( 3, 3), C(, 3), and D(4, ) is drawn in the coordinate plane. The figure is a trapezoid. Use the Distance Formula to find the length of CD. CD perimeter of ABCD: P AB BC CD DA units area of ABCD: A b b h units Draw and classify each polygon with the given vertices. Find the perimeter and area of each polygon. 3. F(, 3), G(, 3), H(, 0) 4. Q( 4, 0), R(, 4), S(, ), T(0, )

8 Chapter 9.4 Perimeter and Area in the Coordinate Plane continued When a figure in a coordinate plane does not have an area formula, another method can be used to find its area. Find the area of the polygon with vertices N( 4, ), P(, 3), Q(4, 3), and R(, ). Step Step Step 3 Draw the polygon and enclose it in a rectangle. Find the area of the rectangle and the areas of the parts of the rectangle that are not included in the figure. rectangle: A bh units units a: A bh 5 5 units b: A bh 6 3 units c: A bh Subtract to find the area of polygon NPQR. A area of rectangle area of parts not included in figure units Find the area of each polygon with the given vertices

9 Chapter 9.6 Geometric Probability The theoretical probability of an event occurring is number of outcomes in the event P. number of outcomes in the sample space The geometric probability of an event occurring is found by determining a ratio of geometric measures such as length or area. Geometric probability is used when an experiment has an infinite number of outcomes. Finding Geometric Probability Use Length A point is chosen randomly on AD. Find the probability that the point is on BD. all points on BD P all points on AD BD AD Use Angle Measures Use the spinner to find the probability of the pointer landing on the 60 space. all points in 60 region P all points in circle A point is chosen randomly on EH. Find the probability of each event.. The point is on FH.. The point is not on EF. 3. The point is on EF or GH. 4. The point is on EG. Use the spinner to find the probability of each event. 5. the pointer landing on the pointer landing on the pointer landing on 90 or the pointer landing on 30

10 Chapter 9.6 Geometric Probability continued You can also use area to find geometric probability. Find the probability that a point chosen randomly inside the rectangle is in the triangle. area of triangle: A bh area of rectangle: A bh all points in triangle P all points in rectangle area of triangle area of rectangle 9 cm 50 cm The probability is P cm 0(5) 50 cm Find the probability that a point chosen randomly inside the rectangle is in each shape. Round to the nearest hundredth. 9. the square 0. the triangle. the circle. the regular pentagon

11 Answers for the chapter Extending Perimeter, Circumference, and Area Chapter 9.. A 60 in. b 7 mm 3. A 9 m 4. h 6 cm 5. P 50 in. 6. b 4 ft 7. A 70 cm 8. d 36 ft 9. d m 0. d 3 mm. d 4x m. A 4a b in Chapter Chapter 9.. C 6 cm. A 5 m 3. A ft 4. C 0 in. 5. C 4 in. 6. r 9 cm 7. A cm 8. A 58. in 9. A 3. m 0. A ft 4. triangle; P 6 units; A units Chapter 9.3. A 450 yd. A 0 mm 3. A ft 4. A 36 m 5. A 97 mm 6. A 35. cm square; P 7.9 units; A 0 units 5. A 9 units 6. A 0.5 units Chapter