A triangle is inscribed in a circle with a radius = 3, what’s the area of the shaded region?
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- เผยแพร่เมื่อ 20 พ.ค. 2024
- How to find the area of a region - triangle inscribed in circle. Learn more math at TCMathAcademy.com/.
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I first found the area of the circle (with π=3.1.416) = 28.2744. Then, the area of both triangles 2(1/2bh = 1/2(9) = 2*1/2 * 9 = 1*9 = 9), then subtracted the triangular areas from the circle (28.2744 - 9 = 19.2744), and subtracting 1/2 of the circle-the half not useful for determining the needed area-1/2(28.2744) = 14.1372: 19.2744 - 14.1372 = 5.1372 = both of the arced areas surrounding the triangles. Finally, I divided this by 2 to get 1/2 the arced areas = 2.5686 for the final answer. Your use of π/2 was way easier and more elegant.
wouldn't this be easier? area of circle would be 9pi, then find the area of the quarter circle with the shading =9pi/4, then subtract the triangle 1/2 *3 *3 and you're left with (9pi/4)-9/2 answer is (9pi-18)/4
That's how I did it.
Same here. He’s complicating it by using a semicircle. Just use the quarter circle and a 3x3 triangle. Saves an extra step of having to divide by 2 again.
@@MM-gz3ou Yes. It's such a weird method and I don't know what the advantage is supposed to be.
@@gavindeane3670And that is what I did as well. I think that is the most logical way to do this, the area of a sector minus the are of the triangle to get the area of the segment of the arc. Anything else is only more confusing IMO
My though process was similar to many in getting the answer. However, it was good to see how others did it and the geometry used.
Solved at the thumbnail, my answer is approximately 2.57 units squared.
First, the area of the full circl is pi * 3"2 = pi * 9 = ~28.27 units squared.
Thus, the area of the quadrant where the shaded area lies is ~7.07 units squared.
Next, the area of the right triangle inscribed in that quadrant is 1/2 (3^2) = 1/2 (9) = 4.5.
Therefore, the area of the shaded region is ~7.07 - 4.5 = ~2.57 units squared.
2.5, derived in my head, 50 seconds timed. Using paper and pencil would have gotten a more precise answer but in 4 times the time.
Pretty simple, but these probs involving funny, entangled rectangles, triangles, and circles are challenging for me so this what I'm working on today.
Thank you Tutor John for your contributions to my math Renaissance! 😀
Calculate the area of the circle: 9π ≈ 28.3. Divide by 4: ≈ 7. Calculate the area of the right triangle: (1/2)*3*3 = 4.5. Subtract the triangle's area from the fractional area of the circle: ≈7 - 4.5 ≈ 2.5.
Find area of circle =πr²
Find area of triange = 3x3
Divide area of circle by 2
Subtract area of triangle
Divide by 2
Answer 2.568
Great lesson!
I knew right away what I needed to do but didn't know the rules of how to do it.
But you took the time to show me us and I am so grateful ❤
I'm gonna give you ona these 👍 and 1a these 😎 4 being so cool and 💪😬👋❤️🌎
Thanks Math Man
Thank you
I did it with the full circle area minus the square area (2 triangles top and bottom make a square) divided by 4 as the blue area is 4 times in the circle.
Area of the circle is 9π. The area if the square is easy with the Pythagorean theorem 3 squared plus 3 squared equals the square of the side of our big square inscribed in the circle. As the surface of the square is the side multiplied by itself it is 9+9=18. 9π-18 gives us the four surface similar to the blue area so we divide by 4. (9π-18)/4
Figure out the area of a 1/4 Circle and subtract the area of the small isosceles triangle .
The area in the shaded region is 2.568 square units
1/4 of circle = πr^2/4 or 9π/4
Area of triangle = 9/2
Hence x = 9π/4 - 9/2
or
x = 9π/4 - 18/4
or
x = 9(π-2)/4
or
If you want an approximation:
x=2.569
The thing that tripped me up is that the triangle didn't look like an isosceles triangle. The left side looked slightly longer than the right side. Once that was clarified, then it becomes straightforward.
If you want a reason one can do the quadrant, it just relies on the properites of circular symmetry
The last comment is the easiest method, i.e. calculate the area of the quarter circle, then subtract the area of ONE triangle.
deduce:
--the two triangles are identical because it's given the circle has radius = 3 and the triangles has side = 3.
--as the bottom line of the two triangles has one leg listed as radius this line is a diameter.
The area desired is 1/4 circle - 1 triangle.
circle:
pi(r^2)
1/4 circle:
= (1/4)(pi)(r^2)
= (1/4)(pi)(3^2)
= (9/4)pi
triangle:
= (1/2)(b)(h)
= (1/2)(3)(3)
= (9/2)
shaded area:
= (9/4)(pi) - (9/2)
= 7.068 - 4.500
= 2.568 units^2
That is how I did it
(9 * pi - 2 * 3* 3) / 4 = A .. should give the answer
This is HOW I would do it ..
The area of the circle is Pi r squared .. Pi 3^2 .. and then we want half that
The area of the two triangles is the same a square 3 x 3
Subtract the area of the two triangles from the area of half the circle .. and then half that.
r=3
Area = ( Pi* r^2 -18 ) / 4
It is a waste of time to talk about the semi-circle, just calculate the area of the quadrant, or 1/4th the area of the circle and go from there. Guess if you want to emphasize similar triangles one can do it your way but its harder with unnecessary steps
3.14×9÷4÷4- .5×3×3=2.565
Why not just start with the formula for the area of a sector, A=θr^2 - area of a triangle? Why make something so easy into something so complicated?
Woo Hoo 2.565 on the button. Area of circle then divide by 2. area of rt tri multiply by 2 . Subtract from semi-circle.
divide by 2 for 1 arc. i think thats what i did. thanks for the fun.
EXACT Answer is 9(π - 2)/4
Without a calculator, it is 4.5pi -9
.0833
Ugh, you did something wrong. I hope you watched the video after giving your answer.
Suggestion: it always helps to get at least a rough estimate by just looking at the problem, and look at your final answer to see if it is anywhere near your estimate. That slice that you're trying to find the area of has a straight side of something more than 3. And at its widest, it appears to be a little less than 1. So the area should be something less than 3, but certainly greater than 1. Your .0833 is nowhere near that.
Mr. TH-cam Mathman, when you use the term "not that difficult", I know it's time to buckle up!
.32 square units?
.65 units
2.565
Sorry 2.565
Can I solve it NO!? and no convoluted explanation and you wonder why people move.on when they see the crazy explanation. I pray for you!!!
2.5685 =area of full circle pir²=28.27433÷4=7.068 -area of rat=bxh÷2 =3×3=9÷2=4.5:- 7.068 -4.5 = 2.5685 =area of remaining semicircle even dumbo can do it easy way! Why bother with all the rest?!
This explanation should have taken 2-4 minutes, rather than 10.
I think the extra time helps, as then people have a short time to think about it before going on the next bit.
I remember at school sometimes a teacher would speed up as they went along. If I asked what happened after a particular point they would just start from the beginning again and speed up again by the time they got to the part the question was about. It happened too often for it not to be deliberate, it used to really piss me off.