What is the area of a triangle with sides measuring 6.8 cm, 6 cm and 8.8 cm? Round to the nearest tenth.

Answer 1

Hi Victoria,

There are a couple ways to solve this. One includes using the law of cosines to find the height of the triangle, then using A = 1/2*b*h. The other uses Heron's formula to find it using the semi-perimeter.

I drew the triangle with 6 on the left, 6.8 on the right, and 8.8 as the base. I called the bottom left angle, which is formed by 8.8 and 6, as "theta". By law of cosines, we get:

6.8² = 6² + 8.8² - 2(6)(8.8)*cos(Θ)

46.24 = 113.44 - 105.6*cos(Θ)

-67.2 = -105.6*cos(Θ)

cos(Θ) = 0.636363

Θ = 50.48 deg

We can draw a perpendicular line segment from the top angle to the base and call it H. We can then state:

sin(Θ) = H / 6

sin(50.48) = H / 6

H = 4.63

So with a base of 8.8 and a height of 4.63, the area is 20.372.

Heron's formula is a bit more straightforward. First find the "semi-perimeter".

s = (a + b + c)/2

s = (6 + 6.8 + 8.8)/2

s = 10.8

Then just plug this into Heron's formula

A = sqrt(s*(s - a)*(s - b)*(s - c))

= sqrt(10.8*(10.8 - 6)*(10.8 - 6.8)*(10.8 - 8.8))

= sqrt(10.8*4.8*4*2)

= sqrt(414.72)

= 20.365

The slight difference in answers is due to rounding errors from the determination of Θ and H. I'm sure if I reported each to 5 or 6 decimals it would've been a much closer match. Both round to 20.4, though, so for the purpose of this question either method would be okay.

Hope this helps!

Answer 2

Dear students and experts. Hero from Alexandria solved this problem many, many years ago. The area of the triangle

A = √p·(p - a)·(p - b)·(p -c)

We call this formula HERON'S formula. Here p - is half of the perimeter of the triangle, a, b, and c measures of the sides of the triangle.

All other solutions - just disturbance of the air.

Good luck to everybody!

Answer 3

The area of a triangle is (1/2)*(base)*(height). To find the height, divide the triangle as shown. Line x is perpendicular to the base, which is 8.80 cm. Each triangle is the same. Take one and use the Pythagorean Theorem to calculate x. I find x, the height, of 4.06 cm.

Then use:

Area - (1/2)*(Base)*(Height)

Area = (1/2)*(8.80)*(4.06)

Area = 17.95 cm²