Categories

# 2D Geometry – Areas related to circle AMC 8 2017 Problem 25

Try this beautiful problem from AMC 8. It involves calculating the area of a sector of a circle. We provide sequential hints so that you can try the problem.

## Competency in Focus: 2D Geometry (Areas related to circle)

This problem from American Mathematics Contest 8 (AMC 8, 2017) is based on calculation of areas related to circle. It is Question no. 25 of the AMC 8 2017 Problem series.

## Next understand the problem

In the figure shown, $\overline{US}$ and $\overline{UT}$ are line segments each of length 2, and $m\angle TUS = 60^\circ$. Arcs $TR$ and $SR$ are each one-sixth of a circle with radius 2. What is the area of the region shown? $\textbf{(A) }3\sqrt{3}-\pi\qquad\textbf{(B) }4\sqrt{3}-\frac{4\pi}{3}\qquad\textbf{(C) }2\sqrt{3}\qquad\textbf{(D) }4\sqrt{3}-\frac{2\pi}{3}\qquad\textbf{(E) }4+\frac{4\pi}{3}$
##### Source of the problem
American Mathematical Contest 2017, AMC 8 Problem 25

### Finding the area of a triangle and sector of a circle. (Area related to circles)

5/10
##### Suggested Book

Do you really need a hint? Try it first!
C0nstruction : Let $X$ and $Y$ are the centres of the scetors $ST$ and $TR$ Now Let us join $SX$ and $TY$ What do you think? Will the points $U,S,\textbf{ and}\quad X$ be in a straightline?
$U,S,\textbf{ and}\quad X$ will be in a straight line because $\angle STU =60^{\circ}$ And angle of a  circle is $360$  i.e., $\angle SXR = \angle TYR = 60^{\circ}$ [Since sector($SXR$)=$\frac{1}{6}circle$] Then $UXY$ will make an equilateral triangle.
So after construction the figure will look like this : Therefore, The required area = Area of $\triangle UXY$ – $2 \times$ Area of the sector $SXR$.
Area of equilateral triangle $\triangle UXY= 4\sqrt{3}$ And the are of sector $SXR= \frac{2\pi}{3}$ ANS : $4\sqrt{3}-\frac{4\pi}{3}$
Area of an equilateral triangle =$\frac{a^2\sqrt{3}}{4}$ [where $a$ is a sied of the triangle] Area of a sector of a circle of angle $\theta$ = $\frac{\theta}{360}\pi r^2$ [where $r$ is the radius of the circle]

## AMC - AIME Program

AMC - AIME - USAMO Boot Camp for brilliant students. Use our exclusive one-on-one plus group class system to prepare for Math Olympiad

## External Tangent | AMC 10A, 2018 | Problem 15

Try this beautiful Problem on geometry based on circle from AMC 10A, 2018. Problem-15. You may use sequential hints to solve the problem.

## Dice Problem | AMC 10A, 2014| Problem No 17

Try this beautiful Problem on Probability from AMC 10A, 2014. Problem-17, You may use sequential hints to solve the problem.

## Problem on Curve | AMC 10A, 2018 | Problem 21

Try this beautiful Problem on Co-ordinate geometry from AMC 10A, 2018. Problem-21, You may use sequential hints to solve the problem.

## Right-angled Triangle | AMC 10A, 2018 | Problem No 16

Try this beautiful Problem on triangle from AMC 10A, 2018. Problem-16. You may use sequential hints to solve the problem.

## Finding Greatest Integer | AMC 10A, 2018 | Problem No 14

Try this beautiful Problem on Algebra from AMC 10A, 2018. Problem-14, You may use sequential hints to solve the problem.

## Length of the crease | AMC 10A, 2018 | Problem No 13

Try this beautiful Problem on triangle from AMC 10A, 2018. Problem-13. You may use sequential hints to solve the problem.

## Right-angled shaped field | AMC 10A, 2018 | Problem No 23

Try this beautiful Problem on triangle from AMC 10A, 2018. Problem-23. You may use sequential hints to solve the problem.

## Area of region | AMC 10B, 2016| Problem No 21

Try this beautiful Problem on Geometry on Circle from AMC 10B, 2016. Problem-20. You may use sequential hints to solve the problem.

## Coin Toss Problem | AMC 10A, 2017| Problem No 18

Try this beautiful Problem on Probability from AMC 10A, 2017. Problem-18, You may use sequential hints to solve the problem.

## GCF & Rectangle | AMC 10A, 2016| Problem No 19

Try this beautiful Problem on Geometry on Rectangle from AMC 10A, 2010. Problem-19. You may use sequential hints to solve the problem.

This site uses Akismet to reduce spam. Learn how your comment data is processed.