How to Calculate the Area of Triangles

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Diego Fri, Mar 25, 2022 06:59 AM UTC

The Height of the triangle is the perpendicular segment from a vertex to the line containing the opposite side. The opposite side is called the Base of the triangle. The terms height and base are also used to represent the segment lengths.

1. Page 1 of 7
8.4 Area of Triangles
Goal The amount of material needed to
Find the area of triangles. make the sail at the right is
determined by the area of the
Key Words triangular sail.
• height of a triangle The height and base of a triangle are
• base of a triangle used to find its area.
The height of a triangle is the perpendicular segment from a vertex
to the line containing the opposite side. The opposite side is called
the base of the triangle . The terms height and base are also used
to represent the segment lengths.
height height height
base base base
In a right triangle, A height can be A height can be
a leg is a height. inside the triangle. outside the triangle.
As shown in Activity 8.4, the area of a triangle is found using a base
and its corresponding height.
AREA OF A TRIANGLE
1
Words Area 5 }} (base)(height)
2
1
Symbols A 5 }}bh
2 b
Triangles with the Same Area Triangles can have the same area
without necessarily being congruent. For example, all of the
triangles below have the same area but they are not congruent.
8 8 8 8
13 13 13 13
8.4 Area of Triangles 431

2. Page 2 of 7
EXAMPLE 1 Find the Area of a Right Triangle
Find the area of the right triangle.
6 cm
Solution
Use the formula for the area of a triangle. 10 cm
Substitute 10 for b and 6 for h.
1
A 5 }}bh Formula for the area of a triangle
2
1
5 }}(10)(6) Substitute 10 for b and 6 for h.
2
5 30 Simplify.
ANSWER © The triangle has an area of 30 square centimeters.
EXAMPLE 2 Find the Area of a Triangle
Find the area of the triangle.
Solution 8 ft
1
A 5 }}bh Formula for the area of a triangle
2
1
5 }}(8)(5) Substitute 8 for b and 5 for h.
2
5 20 Simplify.
ANSWER © The triangle has an area of 20 square feet.
Visualize It! EXAMPLE 3 Find the Height of a Triangle
To help you determine Find the height of the triangle, given that
the base and height of a
tilted triangle, turn your
its area is 39 square inches. h
book so that the base is 13 in.
horizontal.
Solution
1
h A 5 }}bh Formula for the area of a triangle
2
13 1
in. 39 5 }}(13)h Substitute 39 for A and 13 for b.
2
78 5 13h Multiply each side by 2.
65h Divide each side by 13.
ANSWER © The triangle has a height of 6 inches.
432 Chapter 8 Polygons and Area

3. Page 3 of 7
Area of Triangles
In Exercises 1–3, find the area of the triangle.
1. 2. 3.
8 in. 7 yd 16 cm
15 cm
9 in. 12 yd
4. A triangle has an area of 84 square inches and a height of
14 inches. Find the base.
EXAMPLE 4 Areas of Similar Triangles
a. Find the ratio of the areas of the similar B
triangles. 2
E
b. Find the scale factor of T ABC to TDEF C 4 A
and compare it to the ratio of their areas. 3
F 6 D
Solution T ABC S TDEF
1 1
a. Area of T ABC 5 }}bh 5 }}(4)(2) 5 4 square units
2 2
1 1
Area of TDEF 5 }}bh 5 }}(6)(3) 5 9 square units
2 2
Student Help
Area of T ABC 4
LOOK BACK
Ratio of areas 5 }} 5 }}
Area of TDEF 9
To review scale factor, 2
see p. 366. b. The scale factor of T ABC to TDEF is }}.
3
22 4
The ratio of the areas is the square of the scale factor: }}2 5 }}.
3 9
The relationship in Example 4 is generalized for all similar polygons in
the following theorem.
THEOREM 8.3
Areas of Similar Polygons
Words If two polygons are similar with a scale C
a G
factor of }}, then the ratio of their areas B
b a
a2
is }}2 . F
b b
A D
Symbols If ABCD S EFGH with a scale factor
a Area of ABCD a2
of }}, then }} 5 }}2 . E H
b Area of EFGH b
8.4 Area of Triangles 433

4. Page 4 of 7
8.4 Exercises
Guided Practice
Vocabulary Check 1. What are the measures of the base and 13 ft
the height of the shaded triangle at the 5 ft
right?
3 ft 9 ft
Skill Check The triangle has a horizontal base of 15 units and a height of 7 units.
Sketch the triangle and label its base and its height.
2. 3. 4.
Practice and Applications
Extra Practice Area of a Right Triangle Find the area of the right triangle.
See p. 690. 5. 6. 7.
7 cm 6 ft 3 yd 5 yd
12 cm 7 ft
Finding Area In Exercises 8–13, find the area of the triangle.
8. 9. 7 yd 10.
4m
8 mm
4 yd
9m
14 mm
11. 12. 13.
12 in. 9 ft 14 ft
4 cm
12 in.
5 cm
14. You be the Judge In the triangle
Homework Help at the right, Trisha says the base is 15
and the height is 4. Luis says that the
Example 1: Exs. 5–7 base is 5 and the height is 12. Who is 15 12
Example 2: Exs. 8–13 right? Explain your reasoning.
Example 3: Exs. 16–20 4
Example 4: Exs. 26, 27
5
434 Chapter 8 Polygons and Area

5. Page 5 of 7
15. Visualize It! Draw three different triangles that each have an
area of 24 square units.
Using Algebra In Exercises 16–18, A gives the area of the
triangle. Find the missing measure.
16. A 5 22 ft2 17. A 5 63 cm2 18. A 5 80 m2
b
4 ft h b
14 cm
16 m
19. Finding the Height A triangle has an area of 78 square inches and
a base of 13 inches. Find the height.
20. Finding the Base A triangle has an area of 135 square meters and
a height of 9 meters. Find the base.
Tiles In Exercises 21 and 22, use the diagram of the tile pattern.
2 in.
4 in.
21. Find the area of one triangular tile.
22. The tiles are being used to make a rectangular border that is
4 inches high and 48 inches long. How many tiles are needed
for the border? (Hint: Start by finding the area of the border.)
Complex Polygons Find the area of the polygon by using the
triangles and rectangles shown.
23. 24. 25. 8 cm
5 ft
4m
12 ft 6 cm
7 ft
4m
5 ft 5 cm
7m
Areas of Similar Triangles In Exercises 26 and 27, the triangles are
similar. Find the scale factor of TPQR to TXYZ. Then find the ratio of
their areas.
26. P 27. P
3 Y
Y 6
P 5 R
6 P 9 R 8
X 10 Z X 12 Z
8.4 Area of Triangles 435

6. Page 6 of 7
Area of a Regular Octagon In Exercises 28–30, use the regular
octagon at the right.
Geology B C
28. Find the area of TGXF in the octagon.
A D
29. Copy the diagram. To form congruent
X
triangles, connect the following pairs of
vertices: A and E, B and F, C and G, D H E
and H. How many triangles are formed?
G 4 F
30. What is the area of the octagon? Explain.
31. Rock Formations Many basaltic columns are hexagonal. The
top of one of these columns is a regular hexagon as shown
below. Find its area. (Another photograph of basaltic columns
is on page 408.)
ROCK FORMATIONS
Geologists learn about the
structure of the earth by
studying rock formations such 7.8 in.
as the basaltic columns at the
Giant’s Causeway in Ireland
pictured above.
9 in.
Application Links
CLASSZONE.COM
EXAMPLE Using the Pythagorean Theorem
Find the area of the triangle. 13
5
b
Solution
First, find the base. Use the Pythagorean
Theorem to find the value of b.
Student Help (hypotenuse)2 5 (leg)2 1 (leg)2 Pythagorean Theorem
2 2 2
LOOK BACK (13) 5 (5) 1 (b) Substitute.
To review the 2
Pythagorean Theorem, 169 5 25 1 b Simplify.
see pp. 192 and 193.
144 5 b2 Subtract 25 from each side.
12 5 b Find the positive square root.
Use 12 as the base in the formula for the area of a triangle.
1 1
A 5 }}bh 5 }}(12)(5) 5 30 square units
2 2
Using the Pythagorean Theorem Find the area of the triangle.
32. 33. 34.
26
a 20 12
6 10
24
b
b
436 Chapter 8 Polygons and Area

7. Page 7 of 7
Standardized Test 35. Multiple Choice Given that the area of the triangle is 99 square
Practice meters, what is the height of the triangle?
A 5 99 m2
X
A 4.5 m B 9m
X h
X
C 11 m D 22 m
X
22 m
Mixed Review Trapezoids Find the value of x in the trapezoid. (Lesson 6.5)
36. 6 37. 20 38. 18
x 16 21
x x
12
Algebra Skills Naming Coordinates Give the coordinates of the point.
(Skills Review, p. 664)
A y
39. A 40. B
C
41. C 42. D B
1
43. E 44. F 1 x
D E
F
Quiz 2
Find the area of the polygon. (Lessons 8.3, 8.4)
1. 2. 3. 5m
3 in.
7 in. 8m
12 cm 3m
7m
4. 5. 14 mm 6.
12 yd
14 ft
8 mm
16 yd
22 ft
In Exercises 7–9, A gives the area of the polygon. Find the missing
measure. (Lessons 8.3, 8.4)
7. A 5 48 in.2 8. A 5 90 m2 9. A 5 63 cm2
9 cm
6 in. h
b 15 m b
8.4 Area of Triangles 437

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