lesson thirty: ‘round about. inscribed and circumscribed figures we already know a bit about...
TRANSCRIPT
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LESSON THIRTY:
‘ROUND ABOUT
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INSCRIBED AND CIRCUMSCRIBED FIGURES
• We already know a bit about circles.• You know how to find the radius.• You know how to find the diameter.• Using these, we found the circumference and
area.
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INSCRIBED AND CIRCUMSCRIBED FIGURES
• But if I gave you the area or circumference, could you find the radius or diameter?
• In essence, could you do what we’ve been doing…backwards?
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INSCRIBED AND CIRCUMSCRIBED FIGURES
• Doing this is simple algebra.• Up until now, we’ve taken the equations or
and plugged in the radius.• Now, depending on what we’re given, we’ll
plug in the A or the C.
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INSCRIBED AND CIRCUMSCRIBED FIGURES
• Let’s say I told you that the circumference of a certain circle was 12.
• All you must do is use this value for your circumference.
12 =
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INSCRIBED AND CIRCUMSCRIBED FIGURES
• We then divide both sides be in order to isolate r.
12 = becomes
6 =
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INSCRIBED AND CIRCUMSCRIBED FIGURES
• What if I told you that the area of a certain circle was 49.
• All you must do is use this value for your area.
49 = • Firstly, we divide both sides by
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INSCRIBED AND CIRCUMSCRIBED FIGURES
• Once we’ve done that, all we need to do is take the square root of both sides. So this…
49 = becomes
7 = r• Thus, the radius is seven.
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INSCRIBED AND CIRCUMSCRIBED FIGURES
• Now that we have learned this, we can learn a bit more vocabulary about circles.
• For instance, when two circles share a center, we call them concentric circles.
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INSCRIBED AND CIRCUMSCRIBED FIGURES
• When a figure is perfectly inside a circle as in the figure below, we say that it is inscribed in the circle.
• This is to say that each of the figure’s vertices are on the circle.
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INSCRIBED AND CIRCUMSCRIBED FIGURES
• We can also say that the circle is circumscribed around a polygon.
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INSCRIBED AND CIRCUMSCRIBED FIGURES
• You will be required to find and draw an inscribed and circumscribed circle in various polygons.
• How will we do this? Any ideas.
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INSCRIBED AND CIRCUMSCRIBED FIGURES
• Firstly, we must find the center.• In a regular polygon, this is relatively easy.• Simply enough, you draw two or more lines of
symmetry and their point of intersection.
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INSCRIBED AND CIRCUMSCRIBED FIGURES
• To draw a circle that is inscribed, our circle must touch all the sides of the polygon.
• How do we do this?• Well, we simply take the length on of our
apothem and make that the radius of our circle.
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INSCRIBED AND CIRCUMSCRIBED FIGURES
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INSCRIBED AND CIRCUMSCRIBED FIGURES
• When finding the circle that circumscribes the polygon, we use the radius of the polygon as our radius of the circle.
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INSCRIBED AND CIRCUMSCRIBED FIGURES
• For triangles, we must recall some old knowledge.
• We must remember incenters and circumcenters.
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INSCRIBED AND CIRCUMSCRIBED FIGURES
• You will recall that incenters are the intersections of all the angle bisectors of a triangle.
• One is pictured below.
C
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INSCRIBED AND CIRCUMSCRIBED FIGURES
• The circumcenter is the point of concurrence of the perpendicular bisectors of the sides of a triangle.
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INSCRIBED AND CIRCUMSCRIBED FIGURES
• The circumcenter is equidistant from the vertices of the triangle.
• The incenter is equidistant from the sides of the triangle.
• You can use these to construct a circle around the triangle.
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• With the incenter as the circle’s center and the length to the sides as your radius, you can create a circle inscribed in your triangle.
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INSCRIBED AND CIRCUMSCRIBED FIGURES
• With the circumcenter as the circle’s center and the length to the angles as your radius, you can create a circle circumscribed around your triangle.
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INSCRIBED AND CIRCUMSCRIBED FIGURES
• The last thing I’ll ask you to know is how to construct a regular hexagon inscribed in a circle…given only a point and radius.
• The following video will show you step by step.
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INSCRIBED AND CIRCUMSCRIBED FIGURES
• This one isn’t too difficult!• http://
www.mathopenref.com/constinhexagon.html
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• This is a lot of material so we’ll take our time with it!
• Do not panic!