Thread: Area of an Isosceles Trapezoid

1. Area of an Isosceles Trapezoid

How do I find the area of an isosceles trapezoid circumscribed about a circle when I know the lengths of the bases (8 and 18) but do not know the radius of the circle? Thanks.

2. Originally Posted by garrett88
How do I find the area of an isosceles trapezoid circumscribed about a circle when I know the lengths of the bases (8 and 18) but do not know the radius of the circle? Thanks.
You need to find out the height of the trapezoid, and it seems like you are not given any information about the circle, so you are out of luck. Is the area or circumference of the circle given?

3. No, not given. Either leg of the trapezoid is also the hypotenuse of a right triangle that has the height of the trapezoid as one of the two legs of that right triangle. I don't know if there is a theorem for this but it seems that if I draw two tangents to the circle from any exterior point, and I'll choose a vertex of the trapezoid as my exterior point, those two tangents must be of equal length, 4 and 4 from the top vertex and 9 and 9 from the bottom one. If that's true, then the hypotenuse is 9+4 = 13 and the bottom leg is 5, so the other leg (which is also the altitude of the trapezoid) is 12. Thus the area is 1/2 x 12 (8 + 18). or 216. Would that be correct?

4. Originally Posted by garrett88
No, not given. Either leg of the trapezoid is also the hypotenuse of a right triangle that has the height of the trapezoid as one of the two legs of that right triangle. I don't know if there is a theorem for this but it seems that if I draw two tangents to the circle from any exterior point, and I'll choose a vertex of the trapezoid as my exterior point, those two tangents must be of equal length, 4 and 4 from the top vertex and 9 and 9 from the bottom one. If that's true, then the hypotenuse is 9+4 = 13 and the bottom leg is 5, so the other leg (which is also the altitude of the trapezoid) is 12. Thus the area is 1/2 x 12 (8 + 18). or 216. Would that be correct?
Hello,

all your considerations and calculations are correct - except the last one:

$A=\frac{8+18}{2} \cdot 12=13 \cdot 12 = 156$