Answer: Approximately 1336.50301206033 square cm
=======================================================
Explanation:
If we know the three side lengths of a triangle, then we can use Heron's formula to get the area
Heron's formula is
[tex]A = \sqrt{S(S-a)(S-b)(S-c)[/tex]
with S being the semiperimeter, or half the perimeter, so [tex]S = \frac{a+b+c}{2}[/tex]. We add up all three sides and divide by 2.
----------------
The smaller triangle has side lengths of a = 19, b = 40, c = 41. So we get a semiperimeter of S = (a+b+c)/2 = (19+40+41)/2 = 50
Plug this along with the a,b,c values to get an area of...
[tex]A = \sqrt{S(S-a)(S-b)(S-c)}\\\\A = \sqrt{50(50-19)(50-40)(50-41)}\\\\A = \sqrt{139500}\\\\A \approx 373.496987939661\\\\[/tex]
which is approximate
-----------------
Do the same for the larger triangle as well
S = (a+b+c)/2 = (19+180+181)/2 = 190
[tex]A = \sqrt{S(S-a)(S-b)(S-c)}\\\\A = \sqrt{190(190-19)(190-180)(190-181)}\\\\A = \sqrt{2924100}\\\\A = 1710\\\\[/tex]
Interestingly enough, we get an exact area value this time as opposed to some approximation.
-----------------
The difference between the two results is the area of the shaded region
1710 - 373.496987939661 = 1336.50301206033
This is approximate since the first result was approximate.
Answer:
K = 1710 cm^2 - K = 373.497 cm^2 = 1337 cm^2
Step-by-step explanation:
Why don't you just look for the whole area and deduct the non-shaded area?
Total
Sides:
a = 180 cm
b = 181 cm
c = 19 cm
Angles:
A = 83.9744 °
B = 90 °
C = 6.02558 °
Other:
P = 380 cm
s = 190 cm
K = 1710 cm^2
r = 9 cm
R = 90.5 cm
________________________
Non shaded area:
Answer:
Sides:
a = 40 cm
b = 41 cm
c = 19 cm
Angles:
A = 73.519 °
B = 79.3849 °
C = 27.0961 °
Other:
P = 100 cm
s = 50 cm
K = 373.497 cm^2
r = 7.46994 cm
R = 20.8569 cm
where:
A = angle A
B = angle B
C = angle C
a = side a
b = side b
c = side c
P = perimeter
s = semi-perimeter
K = area
r = radius of inscribed circle
R = radius of circumscribed circle
