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Precalculus : Use trigonometric functions to calculate the area of a triangle

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Example Questions

Example Question #1 : Area Of A Triangle

In triangle ABC , $m\angle C = 30^\circ$ , BC = 40 , and AC = 20 . Find the area of the triangle.

Possible Answers:

100

800

400

1600

200

Correct answer:

200

Explanation:

When given the lengths of two sides and the measure of the angle included by the two sides, the area formula is:

$A = \frac {1}{2}ab\sin C \\$

Plugging in the given values we are able to calculate the area.

$A = \frac {1}{2}\cdot 40\cdot 20 \cdot \sin 30^\circ= 200$

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Example Question #2 : Area Of A Triangle

Find the area of this triangle:

Tri area f

Possible Answers:

68 un^ 2

11.81 un^ 2

136 un^ 2

66.97 un^ 2

Correct answer:

68 un^ 2

Explanation:

To find the area, use the formula associated with side, angle, side triangles which states,

$area = \frac{ 1}{2} a b \sin C$

where and are side lengths and is the included angle.

In our case,

$a=8, b=17, C=90^\circ$ .

Plug the values into the area formula and solve.

$area = \frac{ 1}{2} (8 )( 17 ) \cdot \sin( 90 ^ o ) = 68$

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Example Question #1 : Area Of A Triangle

Find the area of this triangle:

Tri area d

Possible Answers:

88.11 un^ 2

53.96 un^ 2

82. 99 un ^ 2

212.13 un^ 2

Correct answer:

82. 99 un ^ 2

Explanation:

Use the area formula to find area that is associated with the side angle side theorem for triangles.

$area= \frac{ 1}{2 } a b \sin C$

where and are side lengths and is the included angle.

Plugging these values into the formula above, we arrive at our final answer.

$area = \frac{ 1}{2} (11) (23 ) \sin (41^o ) \approx 82.99$

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Example Question #1 : Area Of A Triangle

Find the area of this triangle:

Tri area b

Possible Answers:

25.38 un^ 2

28 un^ 2

23.67 un^ 2

11.83 un^ 2

Correct answer:

11.83 un^ 2

Explanation:

To solve, use the formula for area that is associated with the side angle side theorem for triangles,

$area = \frac{ 1}{2} ab \sin C$

where and are side lengths and is the included angle.

Here we are using 25^o and not 90^o since that is the angle between and .

Therefore,

$a=8,b=7,C=25^\circ$ .

Plugging the above values into the area formula we arrive at our final answer.

$area = \frac{ 1}{2} (8)(7) \sin (25^o ) \approx 11.83$

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Example Question #1 : Area Of A Triangle

Find the area of this triangle:

Tri area a

Possible Answers:

48.74 un^ 2

133.91 un^2

48.84 un^2

66.95 un^ 2

Correct answer:

48.74 un^ 2

Explanation:

Find the area using the formula associated the side angle side theorem of a triangle,

$area = \frac{ 1}{2} ab \sin C$

where and are side lengths and is the included angle.

In this particular case,

$a=15, b=19, C=20^\circ$

therefore the area is found to be,

$area = \frac {1}{2} (15)(19) \sin (20^o ) \approx 48.74$ .

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Example Question #2 : Area Of A Triangle

Find the exact area of a triangle with side lengths of , , and .

Possible Answers:

1.5

$\frac{\sqrt{17}}{2}$

$\frac{\sqrt{135}}{8}$

$\frac{\sqrt {135}}{4}$

2.5

Correct answer:

$\frac{\sqrt {135}}{4}$

Explanation:

Use the Heron's Formula:

$s=\frac{a+b+c}{2}$

$A=\sqrt{s(s-a)(s-b)(s-c)}$

Solve for .

$s=\frac{a+b+c}{2}=\frac{2+3+4}{2}=\frac{9}{2}$

Solve for the area.

$A=\sqrt{s(s-a)(s-b)(s-c)}$

$A=\sqrt{\frac{9}{2}\left(\frac{9}{2}-2\right)\left(\frac{9}{2}-3\right)\left(\frac{9}{2}-4\right)}$

$A=\sqrt{\frac{9}{2}\left(\frac{5}{2}\right)\left(\frac{3}{2}\right)\left(\frac{1}{2}\right)}=\sqrt{\frac{135}{16}}=\frac{\sqrt{135}}{4}$

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Example Question #3 : Area Of A Triangle

What is the area of a triangle with side lengths , , and ?

Possible Answers:

$area=4\sqrt{6}$

area=16

$area=16\sqrt{6}$

$area=\sqrt{92}$

area=4

Correct answer:

$area=4\sqrt{6}$

Explanation:

We can solve this question using Heron's Formula. Heron's Formula states that:

The semiperimeter is

$s=\frac{1}{2}(a+b+c)$

where , , are the sides of a triangle.

Then the area is

$area=\sqrt{s(s-a)(s-b)(s-c)}$

So if we plug in

$s=\frac{1}{2}(4+5+7)$

$s=\frac{1}{2}(16)$

s=8

So the area is

$area=\sqrt{8(8-4)(8-5)(8-7)}$

$area=\sqrt{96}$

$area=4\sqrt{6}$

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Example Question #1 : Area Of A Triangle

What is the area of a triangle with sides , , and ?

Possible Answers:

$area=100\sqrt{2}$

area=200

$area=10\sqrt{2}$

area=10

$area=20\sqrt{2}$

Correct answer:

$area=10\sqrt{2}$

Explanation:

We can solve this question using Heron's Formula. Heron's Formula states that:

The semiperimeter is

$s=\frac{1}{2}(a+b+c)$

where , , are the sides of a triangle.

Then the area is

$area=\sqrt{s(s-a)(s-b)(s-c)}$

So if we plug in

$s=\frac{1}{2}(6+5+9)$

$s=\frac{1}{2}(20)$

s=10

So the area is

$area=\sqrt{10(10-6)(10-5)(10-9)}$

$area=\sqrt{200}$

$area=10\sqrt{2}$

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Example Question #1 : Area Of A Triangle

What is the area of a triangle with side lengths of , , and ?

Possible Answers:

area= 5

area= 7.5

area= 12

area=6

area= 7

Correct answer:

area=6

Explanation:

We can solve this question using Heron's Formula. Heron's Formula states that:

The semiperimeter is

$s=\frac{1}{2}(a+b+c)$

where , , are the sides of a triangle.

Then the area is

$area=\sqrt{s(s-a)(s-b)(s-c)}$

So if we plug in

$s=\frac{1}{2}(3+4+5)$

$s=\frac{1}{2}(12)$

s=6

So the area is

$area=\sqrt{6(6-3)(6-4)(6-5)}$

$area=\sqrt{36}$

area=6

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Example Question #5 : Area Of A Triangle

What is the area of a triangle with side lengths , , and ?

Possible Answers:

area=32.5

area=900

area=33

area=30

area=20

Correct answer:

area=30

Explanation:

We can solve this question using Heron's Formula. Heron's Formula states that:

The semiperimeter is

$s=\frac{1}{2}(a+b+c)$

where , , are the sides of a triangle.

Then the area is

$area=\sqrt{s(s-a)(s-b)(s-c)}$

So if we plug in

$s=\frac{1}{2}(5+12+13)$

$s=\frac{1}{2}(30)$

s=15

So the area is

$area=\sqrt{15(15-5)(15-12)(15-13)}$

$area=\sqrt{900}$

area=30

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Precalculus : Use trigonometric functions to calculate the area of a triangle

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1 : Area Of A Triangle

Example Question #2 : Area Of A Triangle

Example Question #1 : Area Of A Triangle

Example Question #1 : Area Of A Triangle

Example Question #1 : Area Of A Triangle

Example Question #2 : Area Of A Triangle

Example Question #3 : Area Of A Triangle

Example Question #1 : Area Of A Triangle

Example Question #1 : Area Of A Triangle

Example Question #5 : Area Of A Triangle

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