In order to determine the area of a non-right triangle, we can use Heron's formula:

$\displaystyle Area =\sqrt{s(s-a)(s-b)(s-c)}$
$\displaystyle s=\frac{a+b+c}{2}$

Using the information from the question, we obtain:
$\displaystyle s=\frac{8+13+9}{2}=15$
$\displaystyle Area =\sqrt{15(15-8)(15-13)(15-9)}=35.5$

In order to determine the area of a non-right triangle, we can use Heron's formula:

$\displaystyle Area =\sqrt{s(s-a)(s-b)(s-c)}$
$\displaystyle s=\frac{a+b+c}{2}$

Using the information from the question, we obtain:
$\displaystyle s=\frac{16+11+19}{2}=23$
$\displaystyle Area =\sqrt{23(23-16)(23-11)(23-19)}=87.9$

The formula to find the area of a triangle is

$\displaystyle \text{Area}=\frac{\text{base }\times\text{height}}{2}$

Substitute in the given values for area and base to solve for the height, $\displaystyle h$:

$\displaystyle 24=\frac{12h}{2}$

$\displaystyle 12h=48$

$\displaystyle h=4\:cm$

The formula to find the area of a triangle is

$\displaystyle \text{Area}=\frac{base\times height}{2}$

Substitute in the given values for the base and the height to find the area.

$\displaystyle \text{Area}=\frac{5a\times 2a}{2}=\frac{10a^2}{2}=5a^2\:units^2$

The formula used to find the area of the triangle is

$\displaystyle \text{Area}=\frac{base\times height}{2}$

Now, we will need to use a trigonometric ratio to find the length of the height. Because of the angle given, we will need to use $\displaystyle \text{cosine}$, because we are looking for the length of the adjacent side, the triangle's height, and we know the length of the hypotenuse of the smaller triangle formed by the height.

Remember that $\displaystyle \text{cosine }\theta=\frac{adjacent}{hypotenuse}$

$\displaystyle \cos(33^{\circ})=\frac{height}{11}$

Now, solve for the height.

$\displaystyle \text{Height}=11\cos(33^{\circ})=9.23$

Now you can find the area of the triangle:

$\displaystyle \text{Area}=\frac{45\times9.23}{2}=207.7\:units^2$

The formula used to find the area of the triangle is

$\displaystyle \text{Area}=\frac{base\times height}{2}$

Now, we will need to use a trigonometric ratio to find the length of the height. Because of the angle given, we will need to use $\displaystyle \text{cosine}$, because we are looking for the length of the adjacent side—the triangle's height—and we know the length of the hypotenuse of the smaller triangle formed by the height.

Remember that $\displaystyle \text{cosine }\theta =\frac{adjacent}{hypotenuse}$

$\displaystyle \cos(20^{\circ})=\frac{height}{5}$

Now, solve for the height.

$\displaystyle \text{Height}=5\cos(20^{\circ})=4.70$

Now you can find the area.

$\displaystyle \text{Area}=\frac{8\times 4.70}{2}=18.8\:units^2$

The formula used to find the area of the triangle is

$\displaystyle \text{Area}=\frac{base\times height}{2}$

Now, we will need to use a trigonometric ratio to find the length of the height. Because of the angle given, we will need to use $\displaystyle \text{cosine}$, because we are looking for the length of the adjacent side, the triangle's height, and we know the length of the hypotenuse of the smaller triangle formed by the height.

Remember that $\displaystyle \text{cosine }\theta=\frac{adjacent}{hypotenuse}$

$\displaystyle \cos(24^{\circ})=\frac{height}{34}$

Now, solve for the height.

$\displaystyle \text{Height}=11\cos(24^{\circ})=10.05$

Now you can find the area.

$\displaystyle \text{Area}=\frac{45\times 10.05}{2}=226.1\:units^2$

The formula used to find the area of the triangle is

$\displaystyle \text{Area}=\frac{base\times height}{2}$

Now, we will need to use a trigonometric ratio to find the length of the height. Because of the angle given, we will need to use $\displaystyle \text{cosine}$, because we are looking for the length of the adjacent side—the triangle's height—and we know the length of the hypotenuse of the smaller triangle formed by the height.

Remember that $\displaystyle \text{cosine }\theta=\frac{adjacent}{hypotenuse}$

$\displaystyle \cos(18^{\circ})=\frac{height}{3}$

Now, solve for the height.

$\displaystyle \text{Height}=3\cos(18^{\circ})=2.85$

Now you can find the area.

$\displaystyle \text{Area}=\frac{7\times 2.85}{2}=10.0\:units^2$

The formula used to find the area of the triangle is

$\displaystyle \text{Area}=\frac{base\times height}{2}$

Now, we will need to use a trigonometric ratio to find the length of the height. Because of the angle given, we will need to use $\displaystyle \text{cosine}$, because we are looking for the length of the adjacent side—the triangle's height—and we know the length of the hypotenuse of the smaller triangle formed by the height.

Remember that $\displaystyle \text{cosine }\theta=\frac{adjacent}{hypotenuse}$

$\displaystyle \cos(24^{\circ})=\frac{height}{14}$

Now, solve for the height.

$\displaystyle \text{Height}=14\cos(24^{\circ})=12.79$

Now you can find the area.

$\displaystyle \text{Area}=\frac{24\times 12.79}{2}=153.5\:units^2$

The formula used to find the area of the triangle is

$\displaystyle \text{Area}=\frac{base\times height}{2}$

Now, we will need to use a trigonometric ratio to find the length of the height. Because of the angle given, we will need to use $\displaystyle \text{sine}$, because we are looking for the height of the triangle, which in this case is the side opposite to the known angle, and we also know the length of the hypotenuse of the smaller triangle formed by the height.

Remember that $\displaystyle \text{sin }\theta=\frac{opposite}{hypotenuse}$

$\displaystyle \sin(55^{\circ})=\frac{height}{12}$

Now, solve for the height.

$\displaystyle \text{Height}=12\sin(55^{\circ})=9.83$

Now you can find the area.

$\displaystyle \text{Area}=\frac{14\times 9.83}{2}=68.8\:units^2$