Problems on the radius of the circumscribing circle, length of an arc, and sector with solutions.

1. Solutions to the THQ
Collegeof EngineeringandComputerStudies,
St. Michael’s College
Iligan City

2. Collegeof EngineeringandComputerStudies, St. Michael’s College, Iligan City
1. The sides of a triangle are 80 cm, 100 cm, and 140 cm.
Determine the radius of the circumscribing circle.
𝑟 =
𝑎𝑏𝑐
4𝐴
; 𝑠 =
80 + 100 + 140
2
𝑐𝑚 = 160 𝑐𝑚
𝑟 =
80 100 140
4 160 160 − 80 160 − 100 160 − 140
𝑐𝑚
=
175 6
6
𝑐𝑚 ≈ 71.44 𝑐𝑚

3. Collegeof EngineeringandComputerStudies, St. Michael’s College, Iligan City
2. A central angle of 136◦ subtends an arc of 28.5 cm.
What is the radius of the circle?
𝑠 = 𝑟𝜃 ⟹ 𝑟 =
𝑠
𝜃
𝜃 = 136°
𝜋
180
=
34𝜋
45
𝑟𝑎𝑑.
Hence,
𝑟 =
𝑠
𝜃
=
28.5 𝑐𝑚
34𝜋
45
≈ 12.01 𝑐𝑚

4. Collegeof EngineeringandComputerStudies, St. Michael’s College, Iligan City
3. Each of the four circles shown in the figure is tangent to the other
three. (a) If the radius of each of the smaller circles is 𝑎, find the
area of the largest circle. (b) If 𝑎 = 2.71 cm what is the area of
the largest circle?

5. Collegeof EngineeringandComputerStudies, St. Michael’s College, Iligan City
Solution: (a)
Triangle OMN is an equilateral triangle.
Hence, ∠𝑃𝑂𝑄 = 30°.
Consider ⊿𝑂𝑃𝑄:
cos 30° =
𝑎
ℎ
⇒ ℎ =
𝑎
cos 30°
.
The radius of the big circle is:
𝑟 = 𝑎 + ℎ = 𝑎 +
𝑎
cos 30°

6. Collegeof EngineeringandComputerStudies, St. Michael’s College, Iligan City
Solution: (b)
The radius of the big circle is:
𝑟 = 𝑎 + ℎ = 𝑎 +
𝑎
cos 30°
If 𝑎 = 2.71 cm.
𝐴 = 𝜋𝑟2
𝐴 = 𝜋 2.71 +
2.71
cos 30°
2
𝑐𝑚.2
𝐴 = 107.12 𝑐𝑚.2

7. Collegeof EngineeringandComputerStudies, St. Michael’s College, Iligan City
4. Each of the four quarter arcs shown in the figure
measures 16 ft. in length. Find the area of the light
shaded region.
𝐴 =
1
2
𝑟𝑠; 4 16 = 2𝜋𝑟 ⟹ 𝑟 =
32
𝜋
𝑓𝑡.
Area of the light shaded region
= Area of the 4 squares – Area of the 4 sectors
= 4
32
𝜋
2
− 4
1
2
32
𝜋
2
sq. ft
=
2048
𝜋2 sq.ft
≈ 207.51 sq. ft