Unknown Length and Area Problems

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New York State Common Core Math Geometry, Module 5, Lesson 10

Worksheets for Geometry

Student Outcomes

  • Students apply their understanding of arc length and area of sectors to solve problems of unknown area and length.

Unknown Length and Area Problems


Opening Exercise

In the following figure, a cylinder is carved out from within another cylinder of the same height; the bases of both cylinders share the same center. a. Sketch a cross section of the figure parallel to the base. b. Mark and label the shorter of the two radii as π‘Ÿ and the longer of the two radii as 𝑠. Show how to calculate the area of the shaded region, and explain the parts of the expression


  1. Find the area of the following annulus.
  2. The larger circle of an annulus has a diameter of 10 cm, and the smaller circle has a diameter of 7.6 cm. What is the area of the annulus?
  3. In the following annulus, the radius of the larger circle is twice the radius of the smaller circle. If the area of the following annulus is 12πœ‹ units2, what is the radius of the larger circle?
  4. An ice cream shop wants to design a super straw to serve with its extra thick milkshakes that is double both the width and thickness of a standard straw. A standard straw is 4 mm in diameter and 0.5 mm thick.
    a. What is the cross-sectional (parallel to the base) area of the new straw (round to the nearest hundredth)?
    b. If the new straw is 10 cm long, what is the maximum volume of milkshake that can be in the straw at one time (round to the nearest hundredth)?
    c. A large milkshake is 32 fl. oz. (approximately 950 mL). If Corbin withdraws the full capacity of a straw 10 times a minute, what is the minimum amount of time that it will take him to drink the milkshake (round to the nearest minute)?
  5. In the circle given, 𝐸𝐷 is the diameter and is perpendicular to chord 𝐢𝐡. 𝐷𝐹 = 8 cm, and 𝐹𝐸 = 2 cm. Find 𝐴𝐢, 𝐡𝐢, π‘šβˆ πΆπ΄π΅, the arc length of 𝐢𝐸𝐡, and the area of sector 𝐢𝐴𝐡 (round to the nearest hundredth, if necessary).
  6. Given circle 𝐴 with ∠𝐡𝐴𝐢 β‰… ∠𝐡𝐴𝐷, find the following (round to the nearest hundredth, if necessary).
    a. π‘šπΆπ·
    b. π‘šπΆπ΅π·
    c. π‘šπ΅πΆπ·
    d. Arc length 𝐢𝐷
    e. Arc length 𝐢𝐡𝐷
    f. Arc length 𝐡𝐢𝐷
    g. Area of sector 𝐢𝐴𝐷
  7. Given circle 𝐴, find the following (round to the nearest hundredth, if necessary).
    a. Circumference of circle 𝐴
    b. Radius of circle 𝐴
    c. Area of sector 𝐢𝐴𝐷
  8. Given circle 𝐴, find the following (round to the nearest hundredth, if necessary).
    a. π‘šβˆ πΆπ΄π·
    b. Area of sector 𝐢𝐷
  9. Find the area of the shaded region (round to the nearest hundredth).
  10. Many large cities are building or have built mega Ferris wheels. One is 600 feet in diameter and has 48 cars each seating up to 20 people. Each time the Ferris wheel turns πœƒ degrees, a car is in a position to load.
    a. How far does a car move with each rotation of πœƒ degrees (round to the nearest whole number)?
    b. What is the value of πœƒ in degrees?
  11. β–³ 𝐴𝐡𝐢 is an equilateral triangle with edge length 20 cm. 𝐷, 𝐸, and 𝐹 are midpoints of the sides. The vertices of the triangle are the centers of the circles creating the arcs shown. Find the following (round to the nearest hundredth).
    a. Area of the sector with center 𝐴
    b. Area of β–³ 𝐴𝐡𝐢
    c. Area of the shaded region
    d. Perimeter of the shaded region
  12. In the figure shown, 𝐴𝐢 = 𝐡𝐹 = 5 cm, 𝐺𝐻 = 2 cm, and π‘šβˆ π»π΄πΌ = 30Β°. Find the area inside the rectangle but outside of the circles (round to the nearest hundredth).
  13. This is a picture of a piece of a mosaic tile. If the radius of each smaller circle is 1 inch, find the area of the red section, the white section, and the blue section (round to the nearest hundredth).

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