How do you use Heron's formula to find the area of a triangle with sides of lengths #7 #, #5 #, and #6 #?

Question

How do you use Heron's formula to find the area of a triangle with sides of lengths #7   #, #5    #, and #6    #?

Autumn Armstrong · Accepted Answer

#A = sqrt(s(s-a)(s-b)(s-c)) = 6sqrt(6) ~~ 14.6969#

Heron's formula tells us that the area #A# of a triangle with sides of length #a#, #b# and #c# is given by the formula: #A = sqrt(s(s-a)(s-b)(s-c))# Where #s = (a+b+c)/2# is the semi-perimeter In our case, let #a=7#, #b=5# and #c=6#. Then #s = (a+b+c)/2 = (7+5+6)/2 = 9# and we find: #A = sqrt(s(s-a)(s-b)(s-c))# #=sqrt(9 * (9-7) * (9-5) * (9-6))# #=sqrt(9*2*4*3)# #= sqrt(36*6) = sqrt(36)*sqrt(6) = 6sqrt(6) ~~ 14.6969#

Sadie Allen · Answer

undefined To use Heron's formula to find the area of a triangle with sides of lengths 7, 5, and 6, you first need to calculate the semi-perimeter of the triangle, denoted as "s", which is given by the formula:

s = (a + b + c) / 2

where "a", "b", and "c" are the lengths of the triangle's sides. In this case, "a" is 7, "b" is 5, and "c" is 6.

Substitute these values into the formula to find the semi-perimeter:

s = (7 + 5 + 6) / 2
s = 18 / 2
s = 9

Next, use Heron's formula to find the area of the triangle, which is given by:

Area = √(s(s - a)(s - b)(s - c))

Substitute the value of "s" and the side lengths into the formula:

Area = √(9(9 - 7)(9 - 5)(9 - 6))

Now, calculate each term within the square root:

Area = √(9(2)(4)(3))

Multiply the terms inside the square root:

Area = √(9 * 2 * 4 * 3)
Area = √(216)

Finally, calculate the square root of 216:

Area ≈ √216 ≈ 14.6969

Therefore, the area of the triangle with side lengths 7, 5, and 6 is approximately 14.6969 square units.