How to find the area of the incircle of a triangle (shortcut first, then proof)

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Күн бұрын

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@bprpmathbasics 14 сағат бұрын

Let's talk about the area of a triangle when we know its sides! Heron's formula (example and proof) kzbin.info/www/bejne/a6vYhmOLnraCnJY

@yurenchu 7 сағат бұрын

The formula for the area of a triangle is A = ½*(b)*h where b is the length of the base, and h is the height of the triangle. It can readily be seen that in this right triangle, b = 12 and h = 5 and hence A = ½*(12)*5 = 30 However, we can calculate the triangle's area also differently, after drawing three line segments, from the incircle's center to each of the triangle's vertices, which divide the triangle into three smaller triangles. Let r be the radius of the incircle, each smaller triangle has then a height of r and a base that corresponds to one of the sides of the original large triangle. The total Area is then the sum of the areas of these three smaller triangles: A = ½*(12)*r + ½*(5)*r + ½*(13)*r = ½*(12+5+13)*r = ½*(30)*r = 15*r This of course must be equal to the afore-calculated area A = 30 : 15*r = 30 ==> r = 2 So the area of the incircle is πr² = π*(2)² = 4π .

@ToeKat922 Сағат бұрын

yeah thats what i would do

@wyattstevens8574 2 күн бұрын

With Heron's formula, you can even figure out the excircle radii. Divide twice the area by a+b+c for the inradius, and by b+c-a, a-b+c, and a+b-c for the exradii of sides a, b, and c respectively.

@cyruschang1904 6 сағат бұрын

5 + 12 - 13 = 2r r = 2 Triangle area = 5(12)/2 = 30 Circle area = (2)(2)π = 4π

@emmeeemm 3 сағат бұрын

This is the first time I've ever found Heron's Formula useful. Thanks for this!

@ratandmonkey2982 10 сағат бұрын

I love this new series you're doing

@aComedicPianist 3 күн бұрын

4:08 Couldn’t you multiply the right side by sin(C) to generalize it to any triangle as well?

@aComedicPianist 3 күн бұрын

Assuming you had only the side lengths a and b and the angle C, you could use the law of cosines at the bottom replacing the side length c as well. Heron’s formula would probably be nicer if you had a nice number for the semiperimeter.

@wyattstevens8574 2 күн бұрын

@@aComedicPianistIn the circumradius video, s= 9/2, so it's not too bad. In a 5-12-13 triangle it's 15.

@michaelstahl1515 2 сағат бұрын

Thanks for your video ! That was a small step I missed .

@paparmar Күн бұрын

@ 0:47, for a right triangle, an equivalent expression is r = (a+b-c)/2. In this case: r = (5+12-13)/2 = 2.

@MrPaulc222 11 сағат бұрын

Normally, I would do (12*5)/2 = 30 Then split into three triangles of areas 6r + 6.5r + 2.5r 15r = 30. r = 2 Area is 4pi un^2. So seeing you introduce a much faster way is just great. Thank you.

@Mediterranean81 3 күн бұрын

r(a+b+c) = ab r = ab/a+b+c = 2 So Area = 4π

@jackychanmaths 9 сағат бұрын

The formula should be simplified to sqrt((s-a)(s-b)(s-c)/s)

@zawatsky 7 сағат бұрын

Подбором равные касательные, двигаясь по часовой стрелке от нижней левой: 10, 10, 3, 3, 2, 2. Последние две перпендикулярные касательные равны радиусу. Ответ: r=2.

@lasalleman6792 4 сағат бұрын

First find area of triangle. A = 1/2 b * h. Area here is : 30. Second , find radius. Use incircle formula: Radius = area of triangle/semi-perimeter. r = 30/15. r = 2. Use formula for area of circle. Turns out area of circle is 12.57.