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# Is there a rule about area of a triangle in a circle?

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Manager
Joined: 10 Apr 2008
Posts: 53
Is there a rule about area of a triangle in a circle? [#permalink]

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19 Apr 2009, 21:27
If a triangle is inscribed in a semi circle, i.e. it is a right angle triangle with diameter as base... is there any rule/formula about finding the area of this triangle?

Also, will all such triangles in a semi circle have the same area? Or the same perimeter?
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Re: Is there a rule about area of a triangle in a circle? [#permalink]

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19 Apr 2009, 22:53
thinkblue wrote:
If a triangle is inscribed in a semi circle, i.e. it is a right angle triangle with diameter as base... is there any rule/formula about finding the area of this triangle?Sorry, there is no formula such to find the area of the triangle. When we make a triangle in a semicircle.. we are sure about the two things..
1) the 90degree angle..but the other two angle can be any pair of angle which sum to 90 degree
2) secondly, we are sure about the hypotenuse...
But to fix a triangle, we need at least 3 criteria..
So, triangle area cannot be fixed.

Also, will all such triangles in a semi circle have the same area? Or the same perimeter?

No, they dont have the same area or same perimeter if inscribed in a semi-circle.
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Joined: 21 Apr 2009
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Location: DC
Re: Is there a rule about area of a triangle in a circle? [#permalink]

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21 Apr 2009, 16:16
3
KUDOS
The response that you've already received correctly states that there is no way to know the area of triangle ABD at this point; you'd need some more information. Frankly, that's probably all you wanted!

However, let me add a couple things to this discussion, just in case.

First, a diagram, just to give these things handles:

Thus, line AB is a diameter of the circle with center O, point D is arbitrarily chosen on the circle, and C is straight "up" from the middle (not rigorous, but you know what I mean).

You are correct in saying that ADB must be a right angle, regardless of where D is chosen.

1) There is no lower bound on the area of triangle ADB. If we "push" point D further and further and further to the right, the triangle gets shorter and shorter, see? Thus, the area of triangle ADB can get arbitrarily close to zero.

2) There is, however, an upper bound on the area of triangle ADB! Namely, $$r^2$$. When point D coincides with point C, we create a nice 45-45-90 triangle, and visual inspection seems to indicate that it has the maximum area. (the formula is easy to work out: the base of the triangle is the diameter, which is 2r, and the height would then equal r. Thus, $$(1/2) * b * h = (1/2) * 2r * r = r^2$$

We can also get that result analytically: Since we're building off of the diameter, the Base of the triangle is fixed. Thus, in order to maximize area, what should you do? Ah, let's maximize the other variable: the Height. And how do you do that? By putting the height right in the middle where it has the most "headroom", so to speak.

Enjoy!
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Joined: 18 Jul 2008
Posts: 970
Re: Is there a rule about area of a triangle in a circle? [#permalink]

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04 May 2009, 13:46
+1 for good explanation.

Liquidhypnotic wrote:
The response that you've already received correctly states that there is no way to know the area of triangle ABD at this point; you'd need some more information. Frankly, that's probably all you wanted!

However, let me add a couple things to this discussion, just in case.

First, a diagram, just to give these things handles:

Thus, line AB is a diameter of the circle with center O, point D is arbitrarily chosen on the circle, and C is straight "up" from the middle (not rigorous, but you know what I mean).

You are correct in saying that ADB must be a right angle, regardless of where D is chosen.

1) There is no lower bound on the area of triangle ADB. If we "push" point D further and further and further to the right, the triangle gets shorter and shorter, see? Thus, the area of triangle ADB can get arbitrarily close to zero.

2) There is, however, an upper bound on the area of triangle ADB! Namely, $$r^2$$. When point D coincides with point C, we create a nice 45-45-90 triangle, and visual inspection seems to indicate that it has the maximum area. (the formula is easy to work out: the base of the triangle is the diameter, which is 2r, and the height would then equal r. Thus, $$(1/2) * b * h = (1/2) * 2r * r = r^2$$

We can also get that result analytically: Since we're building off of the diameter, the Base of the triangle is fixed. Thus, in order to maximize area, what should you do? Ah, let's maximize the other variable: the Height. And how do you do that? By putting the height right in the middle where it has the most "headroom", so to speak.

Enjoy!
Intern
Joined: 17 Mar 2008
Posts: 12
Re: Is there a rule about area of a triangle in a circle? [#permalink]

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10 May 2009, 13:55
whats the best way to find the diameter of a circle with an inscribed triangle if you are given an arc? so lets say an inscribed triangle with an arc of 24 that is approximately 3/4ths of the circumference. any suggestions?
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Re: Is there a rule about area of a triangle in a circle? [#permalink]

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18 May 2009, 14:41
dakhan wrote:
whats the best way to find the diameter of a circle with an inscribed triangle if you are given an arc? so lets say an inscribed triangle with an arc of 24 that is approximately 3/4ths of the circumference. any suggestions?

Why does the inscribed triangle come into play? I don't see how it adds any helpful information. If there is an Arc and the angle/size is know - it is very easy to find the Diameter.

In your example, the Arc is 3/4th of the circumference, so that means the total circumference is 36.

$$2 * \Pi * R = \Pi * D = 36$$, where R is radius and D is diameter

$$D = \frac{36}{\Pi} \approx 11.5$$
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Re: Is there a rule about area of a triangle in a circle? [#permalink]

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23 Jul 2013, 08:48
Area: The number of square units it takes to exactly fill the interior of a triangle.
Usually called "half of base times height", the area of a triangle is given by the formula below.
• A = b*h / 2
Other formula:
• A = P*r / 2
• A = abc / 4R
Where "b" is the length of the base, "a" and "c" the other sides; "h" is the length of the corresponding altitude; "R" is
the Radius of circumscribed circle; "r" is the radius of inscribed circle; "P" is the perimeter

My doubt here is:
"P" Is the perimeter of the triangle or the perimeter of the circle?
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Re: Is there a rule about area of a triangle in a circle?   [#permalink] 23 Jul 2013, 08:48
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