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

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Example Question #21 :Trigonometric Applications

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

Possible Answers:

area=7056

area=80

area=84

area=16

area=87.5

Correct answer:

area=84

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

$s=\frac{1}{2}(7+24+25)$

$s=\frac{1}{2}(56)$

s=28

So the area is

$area=\sqrt{28(28-7)(28-24)(28-25)}$

$area=\sqrt{7056}$

area=84

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Example Question #22 :Trigonometric Applications

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

Possible Answers:

$area=2\sqrt{21}$

$area=21\sqrt{2}$

$area=4\sqrt{21}$

area=21

Correct answer:

$area=2\sqrt{21}$

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

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

$s=\frac{1}{2}(14)$

s=7

So the area is

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

$area=\sqrt{84}$

$area=2\sqrt{21}$

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Example Question #11 :Area Of A Triangle

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

Possible Answers:

area=4

$area=4\sqrt{2}$

$area=2\sqrt{2}$

area=2

area=8

Correct answer:

$area=2\sqrt{2}$

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

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

$s=\frac{1}{2}(8)$

s=4

So the area is

$area=\sqrt{4(4-3)(4-3)(4-2)}$

$area=\sqrt{8}$

$area=2\sqrt{2}$

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Example Question #11 :Area Of A Triangle

What is the area of a triangle if the sides of a triangle are,, and?

Possible Answers:

$area=4\sqrt{5}$

$area=16\sqrt{5}$

$area=4\sqrt{5}$

$area=\sqrt{5}$

$area=\sqrt{15}$

Correct answer:

$area=\sqrt{15}$

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

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

$s=\frac{1}{2}(10)$

s=5

So the area is

$area=\sqrt{5(5-4)(5-2)(5-4)}$

$area=\sqrt{15}$

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Example Question #11 :Area Of A Triangle

The triangular fence of the T-rex in Jurassic park has sides a,b, and c measuring 100m, 110m, and 120m respectively. What is area of its enclosure?

Possible Answers:

A=2,152m^2

A=5,521m^2

A=2,512m^2

A=5,152m^2

A=5,251m^2

Correct answer:

A=5,152m^2

Explanation:

Use Heron's formula to calculate the area:

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

A, B, and C are side lengths and s is calculated by:

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

Calculate s:

$s=\frac{100+110+120}{2}=165$

Plug side lengths and s into Heron's formula:

$\\A=\sqrt{165(165-100)(165-110)(165-120)}\\ \\=\sqrt{165(65)(55)(45)}\\ \\=5,152m^2$

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Example Question #26 :Trigonometric Applications

Find the area of this triangle:

Tri area h

Possible Answers:

10.83 un^2

6.25 un^ 2

12.25 un^2

4.87 un^2

Correct answer:

10.83 un^2

Explanation:

Use Heron's Formula

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

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

and a=5, b=5, c=5

we can findto be:

$s = \frac{ 1}{2} ( 5 + 5 + 5 ) = 7.5$ .

From here, plug in all our known values and solve.

$\\area = \sqrt{ 7.5 ( 7.5 - 5 ) (7.5 - 5 ) ( 7.5 - 5 )} \\ \\= \sqrt{ 7.5 (2.5 )^3 } \approx 10.83$

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Example Question #27 :Trigonometric Applications

Find the area of this triangle:

Tri area g

Possible Answers:

14.83 un^ 2

16.19 un^ 2

20.98 un^ 2

4.69 un^ 2

Correct answer:

20.98 un^ 2

Explanation:

To find the area, use Heron's Formula,

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

where

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

and

a=6, b=7, c=9 .

Here,

$s= \frac{ 1}{2} ( 6 + 7 + 9 ) = \frac{ 1}{2} \cdot 22 = 11$ .

Now plug in all known values and solve.

$\\ area = \sqrt{ 11 (11- 6 ) ( 11 - 7 ) ( 11 - 9 )} \\ \\= \sqrt{ 11 \cdot 5 \cdot 4 \cdot 2 } \approx 20.98$

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Example Question #31 :Trigonometric Applications

Find the area of this triangle:

Tri area e

Possible Answers:

124.56 un^ 2

15.23 un^ 2

38.50 un^ 2

37.67 un^ 2

Correct answer:

38.50 un^ 2

Explanation:

To solve, use Heron's Formula,

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

where,, andare the side lengths and

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

In this particular case,

a=13, b=11, c=7

thus,

$s = \frac{ 1}{2} ( 13 + 11 + 7 ) = \frac{ 1}{2} \cdot 31 = 15.5$ .

Plugging these values into the Heron's Formula we arrive at our answer.

$\\ area = \sqrt{ 15.5 ( 15.5 - 11 ) (15.5 - 13 ) ( 15.5 - 7 )} \\ \\= \sqrt{ 15.5 \cdot 4.5 \cdot 2.5 \cdot 8. 5 } \approx 38.50$

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Example Question #32 :Trigonometric Applications

Find the area of this triangle:

Tri area c

Possible Answers:

36.33 un^ 2

15.19 un^ 2

24 un^2

72.99 un^2

Correct answer:

24 un^2

Explanation:

To find the area, use Heron's Formula

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

where,, andare the side lengths and $s = \frac{ 1}{2} (a + b + c )$ .

In this case,

a=6, b=8, c=10

thus,

$s= \frac{ 1}{2} (6 + 8 +10)= \frac{ 1}{2} \cdot 24 = 12$ .

Now plugging these values into the Heron's Formula, we arrive at our final answer,

$area = \sqrt{12(12-6 ) (12 - 8)(12-10) } = \sqrt{12 \cdot 6 \cdot 4 \cdot 2 } = 24$

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Example Question #33 :Trigonometric Applications

Given a triangle with lengths 4,8, and 10, find the area of the triangle.

Possible Answers:

$A=\sqrt{231}$

$A=\sqrt{22}$

A=11

$A=2\sqrt{10}$

$A=5\sqrt{17}$

Correct answer:

$A=\sqrt{231}$

Explanation:

Use Heron's formula.

First find S using

$S=\frac{a+b+c}{2}=\frac{4+8+10}{2}=11$

Then apply the Heron's area formula:

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

$A=\sqrt{11(11-4)(11-8)(11-10)}=\sqrt{231}$

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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 #21 :Trigonometric Applications

Example Question #22 :Trigonometric Applications

Example Question #11 :Area Of A Triangle

Example Question #11 :Area Of A Triangle

Example Question #11 :Area Of A Triangle

Example Question #26 :Trigonometric Applications

Example Question #27 :Trigonometric Applications

Example Question #31 :Trigonometric Applications

Example Question #32 :Trigonometric Applications

Example Question #33 :Trigonometric Applications

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