A right triangle has an angle of 30 degrees. Find the area of the triangle if the hypotenuse is 8mm.

[@jimwalker1089]

I'm 66 and 4 months ago I had a minor stroke. It took me a month to figure out how to to put my socks on with the heal in the right place. I love your method of teaching. It reminded me of how to use sin and cos from my surveying days. And the the value of the3 4 5 triangle to square things up in the field without a calculator calculator. Ie my raised garden beds are4 by 8 , I laid out my garden using string lines with a6 by 8 square. I love love regaining knowledge though you.🙂

[@1JuniFx]

The easier way and quicker way would be to calculate the base and apply the sin area rule So Let base = x Cos (30⁰) = x/8 x = 8.Cos (30⁰) Then A = 1/2 (8).(8.Cos (30⁰)).Sin (30⁰) A = 8 sqrt[3] ~ 13.86mm²

[@TheSimCaptain]

This particular triangle is exactly half an equilateral triangle where all the angles are 60° and all the sides are the same length. Therefore an equilateral triangle split down the middle would make a right triangle with the short side equal to half the longest side.

[@tassiedevil2200]

I think this is the insightful point. It's nice if one recognises the 30-60-90 and recall its proportions but really just need this insight and Pythagoras to get the other side-lengths from scratch.

[@mikechappell4156]

@@tassiedevil2200 If you haven't had geometry in 40 years the odds of remembering the relationship of the sides in the 30°,60°,90° triangle are pretty slim unless you actually use geometry/trig for work. I did have a vague recollection of sin 30° being 0.5, but definitely like the idea of doubling the triangle to get an equilateral triangle better. I will probably remember a trick like that for a few years.

[@TheSimCaptain]

@@mikechappell4156 Yep, I think it's because it puts a picture in your mind and it gets you to think in a visual way.

[@YakubibnEsau]

30-60-90 degree triangle. Smallest side is x, hypotenuse is 2x, big leg is xsqrt3. So hypotenuse is 8, small leg 4, big leg is 4sqrt3. Area is 1/2bh, so 1/2x4x4sqrt3. So you have area is 8sqrt3 or about 13.8564 mm^2.

[@BrandurDavidsen-wd2lp]

Cos 30° = s/8 > s = 8 cos 30= 6,9 Sin 30° = e/8 > e= 8 sin 30°= 4 A = 13,8 mm^2

[@nc1609]

Got 13.85 by Sin(30)*8mm=4mm -> 4mm/Tan(30)= 6.93 (Rounded Tan(30) to 0.577)) -> (6.93*4)/2=13.85

[@ManojkantSamal]

*=read as square roor ^= read as to the power Area = 8×(*3)......may be In the given triangle h=8 Sin 30 =(p/h)=(p/8) So, (P/8)=(1/2) 2p=8 P=4 b=*(h^2-p^2) =*{(8^2)-(4^2)} =*{64-16}=*48 =(4×*3) Area =(1/2){p×b} =(1/2){4×4×*3) =(8×*3)mm^2

[@gavindeane3670]

Base = 8 cos 30° = (8√3)/2 = 4√3 Height = 8 sin 30° = 8/2 = 4 Area = ½ × base × height = ½ × 4√3 × 4 = 8√3 mm² 8√3 can be approximated to taste.

[@supriyomondal6900]

Draw a similar triangle and add both the triangle such that (both base are same side) we get each angle 60° . Therefore we get an equilateral triangle , so all sides are same = 8 mm, now apply formula of equilateral triangle √3/4×side^2 , and we get 16√3 which is again divided by 2 that is 8√3 , and you get the answer.

[@mauriziograndi1750]

13.85 squared cm, I just started with the cos 30degrees 0.866x8=6.93 adjacent side and from there Pythagoras for the third side then the area.

[@jonathanjose6531]

30-60-90 triangle properties(leg length) that's the easiest way to, do it. The shortest leg is half of the hypotenuse, the other leg is a root 3

[@junefreeman3788]

how I would have solved it too..

[@sudhangshubhattacharya4991]

Sides of the triangle 8,4 ,√48 ,therefore area 1/2{√48×4)=8√3

[@cyruschang1904]

It's a 30° - 60° - 90° triangle The three sides are 4 mm, 4✓3 mm, 8 mm area is 4 mm x 4✓3 mm ÷ 2 = 8✓3 mm^2

[@PrithwirajSen-nj6qq]

Another method The triangle is ABC AB is base. Rt angle is at B From B we draw a line to the hypotenuse to cut it at point D so that angle ABD is 30 degrees. Then in triangle ABD, AD=BD In triangle BCD angle CBD=angle BCD=60 degrees DC =BD Hence AD =DC =8/2=4 Again in triangle BCD ,we see ang CBD =ang BCD =60 deg The third angle BDC is 60 deg Hence BCB is an equilateral 🔺 Then DC =BD =BC =4 As we get H =8 P=4 Then B=√(64-16)=√48=4√3 Area =1/2*4√3*4 =8√3

[@actuarialscience2283]

30:60:90=1:2:3. 8/3×2=5.333, 8/3×1=2.667. Area=0.5×5.3×2.7=7.1

[@johnhanson6039]

Yes but all of this works because sin 30 is 1/2, so this special depends upon trig. Can memorize special triangles, but better is to understand the underlying trig, so can solve any situation not just small restricted set

[@PrithwirajSen-nj6qq]

Time slot 6-40 to 6-44 U said the ratio of sides opposite to 30 deg ঃ opposite side to 60 degree ঃ opposite side to 90 deg = 1ঃ √3ঃ 2 Then hypotenuse is 8 Perpendicular is 8/2=4 Base 4√3 Area = 1/2*4√3*4 = 8√3 sq cms

[@panlomito]

Area of triangle = ½ x b (base) x h (height) sin (30°) = 1/2 = h / 8 mm so h = 4 mm cos (30°) = V3 / 2 = b / 8 mm so b = 4V3 mm A = ½ x 4V3 x 4 = 8V3 mm² ( ~ 13,9 mm² )

[@debasismajumdar8634]

This happens to be the most easiest way and orally can be done by sin30 and cos30. Maximum time needed 45 sec. No need to get such long useless lecture.

[@esakkyramasamy4649]

In a triangle if the angles are 30 60 n 90 the sides opposite are 1 root3 n two

[@StephenJWalter]

Area to 2 decimal places is 13.86, not 13.85

[@gavindeane3670]

He seems to be very keen on approximating by truncating rather than rounding. I have no idea why.

[@dazartingstall6680]

opposite = sin30 × 8 = 0.5 × 8 = 4 mm adjascent = 4 (√3) a = 0.5 × 4 × 4(√3) a = 2 × 4(√3) a = 8(√3) a = 13.856... mm²

[@dazartingstall6680]

So what happened was... I was just about to do cos30 × 8 to get the third side, so I could do ½bh when I took actual note of the angles involved. So I switched methods mid-stream but forgot to delete the old workin's. And that, for anyone interested, is why there's an unneeded trig calculation at the top.

[@evelynsingh9866]

Explain in more detail how to calculate the base if it is not given,Thanks

[@dazartingstall6680]

(I'm assuming that by "base" you mean, in this diagram, the horizontal side.) Two methods: *Method 1:* cos30 = base/hypotenuse ∴ base = cos30 × 8 *Method 2:* In a 30°, 60°, 90° triangle, the shortest leg is ½ the length of the hypotenuse, and the second-shortest is the length of the shortest leg × √3 Thus, since the hypotenuse is 8 mm, the shortest leg is 4 mm and the other is 4(√3)

[@adgf1x]

ar=8(3^0.5) sq

[@raya.pawley3563]

Thank you

[@wasimahmad-t6c]

12.8

[@russelllomando8460]

got 8 sr 3 30-60-90 has specific rules X - X sr 3 - 2X thanks for the fun

[@gunasekarangovind4021]

Verbose

[@thenetsurferboy]

I have only ever heard of a right angled triangle

[@norbertopanes7262]

A=1/2 x (4)(7) = 14mm Ans.

[@tassiedevil2200]

@norbertopanes7262 You would be right if 7 was the square root of 48. Was it intended as an estimate?

[@ramesankd1604]

3-'14r2h

[@argonwheatbelly637]

8√3 in about 15 seconds in my head. Sorry, I'm a bit tired.

[@danielfitzpatrick5799]

What is he using on his thigh. What is the instrument called

[@danmike2305]

👍

[@ranjeetkumar2940]

Such a lengthy prologue & solution may take months if not ears. Geat nonsense?

[@syedmdabid7191]

Hoc est area= 16√3 sq. unis, respondeo. 😅😂😊😅😢😂

[@mintusaren895]

Are you aware of your country's labour law.

[@garykiltau2124]

Do you ever get to the point? Every one of these you ramble on and on.