Lesson Worksheet: Circumscribed Polygons Mathematics

In this worksheet, we will practice identifying circumscribed polygons and using their properties to find areas or missing lengths.


True or False: In a circle, all of the inscribed angles that intercept the same arc are equal.

  • ATrue
  • BFalse


In the shown figure, 𝐴𝑍=7.5cm, 𝐡𝑍=14.5cm, and 𝐴𝐢=20cm. Given that all sides of triangle 𝐴𝐡𝐢 are tangent to the shown circle, determine the length of 𝐡𝐢.


Find the perimeter of the △𝐴𝐡𝐢.


Find the lengths of 𝐷𝐡 and 𝐸𝐢.

  • A36.58 cm, 45.41 cm
  • B31.66 cm, 49.32 cm
  • C41.49 cm, 49.32 cm
  • D31.66 cm, 41.49 cm


In the following figure, find the perimeter of the square 𝐿𝑀𝑁𝑂 given that the length of the circle’s radius =3cm.


Which of the following diagrams shows a circumscribed polygon?

  • AC
  • BB
  • CD
  • DA


A regular pentagon is circumscribed around a circle of radius 2 cm. Find the length of one side of the pentagon. Give your answer correct to one decimal place.


In the figure, π‘Ÿ=50cm, 𝐴𝐡=97cm, and 𝐢𝐷=109cm. Find the perimeter and the area of 𝐴𝐡𝐢𝐷.

  • A206 cm, 7,857 cm2
  • B412 cm, 7,857 cm2
  • C206 cm, 10,300 cm2
  • D412 cm, 10,300 cm2


Given that 𝐴𝐡𝐢 is an equilateral triangle whose sides touch circle 𝑀 and that the coordinates of point 𝐡 are (βˆ’7,0), determine the equation of circle 𝑀.

  • Aο€Όπ‘₯+72+𝑦+7√36=4912
  • Bο€Όπ‘₯βˆ’72+ο€Ώπ‘¦βˆ’7√36=4912
  • Cο€Όπ‘₯+72+𝑦+7√32=494
  • Dο€Όπ‘₯+72+𝑦+7√33=493


The circle center 𝑃 is inscribed in the equilateral triangle 𝐿𝑀𝑁 of side 63. What is the circumference of the circle?

  • A42√3
  • B21√3
  • C21πœ‹βˆš3
  • D21πœ‹
  • E42πœ‹βˆš3


In a triangle π‘‹π‘Œπ‘, π‘‹π‘Œ=5cm, 𝑋𝑍=3cm, and π‘Œπ‘=4cm. A circle with center 𝑀 is drawn so that 𝑀𝑍 is a radius and π‘‹π‘Œ is bisected at 𝑀. Find the length of the diameter.


In the shown figure, 𝐴𝑍=7.5cm, 𝐡𝑍=13.5cm, and 𝐴𝐢=19cm. Given that all sides of triangle 𝐴𝐡𝐢 are tangent to the shown circle, determine the length of 𝐡𝐢.


Circle 𝑀 is inscribed in △𝐴𝐡𝐢. Determine the triangle’s area.

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