What is the area of a triangle with side lengths 18, 24, and 30?

216
270
244
192

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Good observation that the triangle is similar to a favorite $3-4-5$ right triangle.

We can calculate that the area of the $3-4-5$ triangle is 6. What would the area of this similar triangle be?

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Since the lengths of the triangle have a GCF not equal to $1$ then we can reduce it by the principle of similar triangles, this would give us the "simplest" lengths of $(9, 12, 15)$ . There are several approaches you can work on, here are some:

(1) Since the lengths satisfy the pythagorean relation, $a^{2} + b^{2} = c^{2}$ , we can proceed to $A = (1/2)(bh)$ with the two shorter sides be base or height.

$A = (1/2)(9)(12)$ $A = 54$ and since the side ratio of the original triangle and the simplest triangle, the ratios of their areas is equal to its square, thus,

$A = 54(4) = 216$

(2) Since we were given lengths of sides of the triangle, we can use Heron's Formula: $A = \sqrt{(s)(s - a)(s - b)(s - c)}$ where $s$ is the semi perimeter of the triangle.

$s = (1/2)(18 + 24+ 30)$

$s = 9 + 12 + 15$

$s = 36$

Thus,

$A = \sqrt{(36)(36 - 18)(36 - 24)(36 - 30)}$

$A = \sqrt{6^{6}}$

$A = 216$

Which agrees with (1).

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I committed some errors, on $(9,12, 15)$ sorry about that.

Daniel Cabrales
- 7 years, 9 months ago

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we can do it by either the herons formula or the simple formula... both ways, the area is 216

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Clearly $24^2+18^2=30^2$

Hence it is a Right-angled Triangle

So, Area = $\frac{1}{2}\times base\times height = \frac{1}{2}\times 18\times 24 = 216$

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by heorn's formula :_ [s(s-a)(s-b)(s-c)]^1/2 we get

semi perimeter (S):_ a+b+c/2 (where a,b,c are lengths of sides of triangle)

30+18+24/2= 36

by the heron's formula :- put the value of s 'a'b'c

area of triangle is :_ [36(36-18)(36-24)(36-30)]^1/2 = 216

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**
Heron's Formula
**
.
Let a=18 , b=24 , c=30,
S = a+b+c / 2 = 18+24+30 / 2 = 36
Area = \sqrt{s(s-a)(s-b)(s-c)}
=\sqrt{36(36-18)(36-24)(36-30)}
=\sqrt{36 * 18 * 12 * 6}
=\sqrt{46656}
=216

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From the side lengths 18, 24, and 30, we can see that it forms the ratio of 3:4:5

Hence, we can deduce that this is a right angled triangle, with 18 and 24 forming the right angle. To find the area, we do (18*24)/2=216 and we get the answer.