Areas of Parallelograms and Triangles - · PDF fileIf your answer is not an integer, ... 6. To start, find the height of the trapezoid using the Pythagorean Theorem, a2 + b2 =

Name Class Date

10-1 Practice

Find the area of each parallelogram.

1. 2.

3. 4.

Find the value of h for each parallelogram.

5. To start, write the area formula for a parallelogram. Substitute 12 for b and 4 for h.

A = bh

=

6. 7.

8. The area of a triangle is 36 m2 and the height is 9 m. Find the length of the corresponding base.

9. Algebra In a parallelogram, a base and a corresponding height are in the ratio 5 : 2. The area is 250 cm2. Find the lengths of the base and the corresponding height. (Hint: Use 5x for the base and 2x for the height of the parallelogram.)

10. A triangle has area 16 m2. List all the possible positive integers that could represent the lengths of its base and height.

11. A classmate drew a rectangle with a height of 8 units and a base of 10 units. What is the area of each figure formed when the rectangle is divided along one of its diagonals?

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5

Areas of Parallelograms and Triangles

Form K

Name Class Date

10-1 Practice (continued) Form K

Find the area of each triangle.

12. To start, write the area formula for a triangle. Find b and h in the diagram.

A = 1

2bh

The height h is perpendicular to the base b

so h = 5 and b = 13. 14.

15. 16.

17. Reasoning A parallelogram has sides that are 30 in. and 12 in. long. The length of the height corresponding to the 30-in. base is 8 in. What is the length of the height corresponding to the 12-in. base?

Coordinate Geometry Find the area of a polygon with the given vertices.

18. A(−3, 1), B(−3, 4), C(7, 1), D(7, 4) 19. A(−1, 1), B(−1, 6), C(2, 6)

20. A(2, 2), B(5, 5), C(5, 0), D (2, −3) 21. A(−5, −2), B(−3, 0), C(−3, −4)

Find the area of each figure.

22. 23.

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6

Areas of Parallelograms and Triangles

? + ? = ? .

Name Class Date

10-2 Practice Form K

Find the area of each trapezoid.

1. To start, write the formula for the area of a trapezoid. Find h, b1, and b2 in the diagram.

1 21 ( )2

A h b b= +

The height h is perpendicular to the bases, so

2. 3.

4. Find the area of a trapezoid with bases 16 in. and 7 in. and height 9 in.

5. Find the area of a trapezoid with bases 5 ft and 11 ft and height 7.4 ft.

Find the area of each trapezoid. If your answer is not an integer, leave it in simplest radical form.

6. To start, find the height of the trapezoid using the Pythagorean Theorem, a2 + b2 = c2.

The height is the other leg of the right triangle with leg 16 mm and hypotenuse 20 mm.

h2 + 162 = 202

h2 + 256 = 400

h2 =

h =

7. 8.

Find the area of each trapezoid to the nearest tenth.

9. 10.

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15

Areas of Trapezoids, Rhombuses, and Kites

1 2? , ? , and ? .h b b= = =

??

Name Class Date


Find the area of each kite.

11. To start, write the formula for the area of a kite. Find the lengths of the two diagonals.

1 212

A d d=

d1 = 6 + 6 =

2 ? ? ?d = + =

12. 13.

14. Algebra The diagonals of a kite are in the ratio 3 : 2. The area of the kite is 27 cm2. Find the length of both diagonals. (Hint: Let the lengths of the diagonals be 3x and 2x.)

Find the area of each rhombus.

15. To start, write the formula for the area of a rhombus. Find the lengths of the two diagonals.

1 212

A d d=

16. 17.

Find the area of each rhombus. Leave your answer in simplest radical form.

18. 19. 20.


16

Areas of Trapezoids, Rhombuses, and Kites

?

1 ? 2 ?d = ⋅ =

2 ? 2 ?d = ⋅ =

Name Class Date


Each regular polygon has radii and apothem as shown. Find the measure of each numbered angle.

1. 2. 3.

Find the area of each regular polygon with the given apothem a and side length s.

4. pentagon, a = 1.6 in., s = 2.4 in. To start, write the area formula for a regular polygon. Find the perimeter p of the polygon.

5. hexagon, a = 2.6 ft, s = 3 ft 6. decagon, a = 3.8 in., s = 2.5 in.

7. octagon, a = 9.3 cm, s = 7.7 cm 8. nonagon, a = 5.6 mm, s = 4.1 mm

Find the area of each regular polygon. Round your answer to the nearest tenth.

9. To start, write the area formula for a regular polygon. Find the length of the apothem.

12

A ap=

The radii form three 120° angles at the center. You can use a 30°-60°-90° triangle to find the apothem.

a = shorter leg =

10. 11. 12.


25

Areas of Regular Polygons

12

A ap=

5 ? = ?p ns= = ⋅

longer leg 4.5 = = ?3 3

Name Class Date


Find the measures of the angles formed by (a) two consecutive radii and (b) a radius and a side of the given regular polygon.

13. decagon 14. 12-gon 15. 18-gon

Find the area of each regular polygon with the given radius or apothem. If your answer is not an integer, leave it in simplest radical form.

16. 17. 18.

19. 20. 21.

22. Your friend printed a picture of a regular 18-gon. She wants to cut the 18-gon into right triangles. If she divides the figure into 36 right triangles, what are the measures of the non-right angles of each triangle?

23. A math teacher draws an equilateral triangle with radius 6 in. and a square with the same radius. Which figure has a greater area? To the nearest tenth, how much greater is the area?

24. A family wants to put the tiles shown at the right in their bathroom. Each tile is a regular hexagon with a radius of 1 in. They need to cover an area that is 48 square ft. About how many tiles do they need? Round to the nearest whole tile.

25. An equilateral triangle has a perimeter of 18 cm. Find its area to the nearest tenth.

26. The logo for a company is a regular hexagon inscribed inside a circle. The logo will be painted on the side of the company’s office building. The radius of the circle will be 8 ft. Find the area of the hexagon to the nearest whole foot.


26

Areas of Regular Polygons

Name Class Date


The figures in each pair are similar. Compare the first figure to the second. Give the ratio of the perimeters and the ratio of the areas.

1. To start, find the scale factor. The scale factor is the ratio of the lengths of corresponding sides.

scale factor 1512

=

2. 3. 4.

The figures in each pair are similar. The area of one figure is given. Find the area of the other figure to the nearest whole number.

5. To start, find the scale factor. Then find the ratio of the areas.

16 scale factor 28

=

Area of smaller triangle = 54 in.2

2

2

4area ratio 7

=

6. 7.

Area of larger pentagon = 100 cm2 Area of larger rhombus = 500 m2

The scale factor of two similar polygons is given. Find the ratio of their perimeters and the ratio of their areas.

8. 9:3 9. 10:4 10. 812

11. 420


35

Perimeters and Areas of Similar Figures

Name Class Date

Find the scale factor and the ratio of perimeters for each pair of similar figures.

12. two regular pentagons with areas 50 in.2 and 162 in.2

13. two rectangles with areas 8 m2 and 98 m2

14. two regular pentagons with areas 45 ft2 and 20 ft2

15. two equilateral triangles with areas 8 3 cm2 and 128 3 cm2

16. two circles with areas 27π in.2 and 48π in.2

17. The area of a regular octagon is 120 ft2. What is the area of a regular octagon with

sides 14

the length of the sides of the larger octagon?

18. It takes 12 h to paint a 10-ft-by-18-ft mural. At this rate, how long will it take the same number of people to paint the same mural on a 15-ft-by-27-ft wall?

19. It costs $3.95 to print a 12-in.-by-16-in. color page. A friend needs to print a poster that is twice as long and twice as wide. At this rate, how much would he expect to pay for the poster?

20. The longer base of a trapezoid is 8 ft. The longer base of a similar trapezoid is 13 ft. The area of the smaller trapezoid is 240 ft2. What is the area of the larger trapezoid?

21. Two similar parallelograms have areas 72 m2 and 32 m2. The height of the larger parallelogram is 12 m. What are the lengths of the bases of both parallelograms?

22. Reasoning How would increasing the length and width of a rectangle by a scale factor of 4 affect the area of the new figure? Explain.

23. Error Analysis A parking garage is expanding. The old garage can hold 300 cars. The new garage will hold 900 cars. The old garage has an area of 21,000 ft2. A classmate says the new garage will have an area of 189,000 ft2. Explain the error your classmate has made and give the correct area for the new garage.


36

Practice (continued) Form K

Perimeters and Areas of Similar Figures 10-4

Name Class Date


Find the area of each regular polygon. Round your answers to the nearest tenth.

1. pentagon with side length 8 in. To start, find the measure of a central angle. Then use trigonometry to find a.

The measure of the central angle of the regular

pentagon is = so, · = .

tan 36° = 4 4 4; = = = ? tan 36 ? °

aa

2. hexagon with side length 9 m 3. decagon with radius 4 mm

4. octagon with radius 10 cm 5. 20-gon with radius 5 in.

6. 15-gon with perimeter 90 ft 7. 18-gon with perimeter 126 m

Find the area of each triangle. Round your answers to the nearest tenth.

8. To start, write the SAS formula, then substitute given values into the formula.

(sin 92°)

9. 10.

11. 12.


45

Trigonometry and Area

3605

12

m C∠ =

1 (sin )2

A bc A=

12

=

Name Class Date


13. ABCDEFGH is a regular octagon with center X and radius 5 cm. Find each measure to the nearest tenth. a. m∠FXE b. m∠YXE

c. XY d. FE

e. perimeter of ABCDEFGH f. area of ABCDEFGH

Find the perimeter and area of each regular polygon to the nearest tenth.

14. 15. 16.

17. The Rileys are replacing the carpet on their outdoor deck. The deck is a regular hexagon with radius 9 ft. The cost for carpet and installation is $3.75/ft2. What will it cost to replace the carpet?

The polygons are regular polygons. Find the area of the shaded region to the nearest tenth.

18. 19. 20.

21. Which has the greater area, a regular octagon with side length 10 ft, or a regular nonagon with side length 10 ft? Justify your answer.

22. Several streets intersect to form triangles near Dupont Circle in Washington, D.C. One such triangle is formed by New Hampshire Avenue, Massachusetts Avenue, and 16th Street. The section of New Hampshire Avenue is about 3100 ft long. The section of 16th Street is about 3500 ft long. The angle enclosed by the two streets has a measure of about 35. What is the area of this triangle, to the nearest 100 ft2?


46

Trigonometry and Area

Name Class Date


Name the following in A.

1. the minor arcs

2. the major arcs

3. the semicircles

Find the measure of each arc in R.

4.

To start, identify the central angle that intercepts . The central angle that intercepts has a measure of .

5. 6. 7.

8. 9. 10.

11. 12. 13.

14. 15. 16.

Find each indicated measure for D.

17. m∠EDI 18.

19. 20. m∠IDH

21. 22.

Algebra Find the value of each variable.

23. 24.


55

Circles and Arcs

Name Class Date

10-6

Practice (continued) Form K

Find the circumference of each circle. Leave your answer in terms of π .

25. To start, substitute values into the formula for the circumference of a circle.

C = 2πr = 2π � = m

26. 27. 28.

Find the length of each darkened arc. Leave your answer in terms of π .

29. To start, find the ratio for the measure of the arc to 360.

30. 31. 32.

33. A motorcycle has tires with a 25-in. diameter. A monster truck has tires with a 66-in. diameter. To the nearest inch, how much farther does a monster truck tire travel in one revolution than a motorcycle tire?

34. Hands of a clock suggest an angle whose measure is continually changing. How many degrees does a minute hand move through during each time interval?

a. 9 min b. 35 min c. 45 min

35. A bicycle tire has a diameter of 21 in. How many inches will the bike travel after 100 revolutions of the wheel? Round your answer to the nearest inch.


56

Circles and Arcs

Circles and Arcs

20360

=

Name Class Date


Find the area of each circle. Leave your answer in terms of π .

1. To start, find the radius. Then use the correct area formula.

r = 8 ÷ 2 = ; A = πr2 = π ·

2. 3. 4.

5. Jerry has a lawn sprinkler that sprays water out into a circle. The diameter of the circle is 10 ft. What area can Jerry water with the sprinkler? Round to the nearest tenth.

6. A dog is on a leash that is attached to a pole in the ground. If the leash is 8 ft long, in how much area can the dog move around? Round to the nearest tenth.

Find the area of each shaded sector of a circle. Leave your answer in terms of π .

7. To start, find the ratio of the measure of the arc to 360.

8. 9. 10.

Find the area of sector QRS in R using the given information. Leave your answer in terms of π.

11. r = 4 in., = 135 12. r = 10 cm, = 90


65

Areas of Circles and Sectors

90360

=

Name Class Date


13. 14.

15. 16.

Find the area of the shaded region. Leave your answer in terms of π and in simplest radical form.

17. 18.

19. 20.

21. The school is planting a circular garden. If the diameter of the garden is 6 ft, what is the garden’s area to the nearest tenth of a square foot?

22. An apartment complex is replanting grass seed in a play area. The area is a circle that is 30 ft in diameter. They need 1 lb of grass seed for every 300 square ft. Find the area to the nearest tenth of a square foot. Then calculate how many 1-lb bags of grass seed they need.

23. A circular fountain is surrounded by flowers. The fountain’s diameter is 8 ft. The flowers extend 1.5 ft out from the fountain on all sides. What area do the flowers cover to the nearest tenth of a square foot?


66

Areas of Circles and Sectors

Find the area of each shaded segment. Round your answer to the nearest tenth.

Documents

Areas of Parallelograms and Triangles - · PDF fileIf your answer is not an integer, ... 6. To start, find the height of the trapezoid using the Pythagorean Theorem, a2 + b2 =