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MATH / Three semicircles of radius 1 are constructed on diameter (AB)¯ of a semicircle of radius 2. The centers of the small semicircles divide (AB)¯ into four line segments of equal leng…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

Three semicircles of radius 1 are constructed on diameter AB¯\overline{AB} of a semicircle of radius 2. The centers of the small semicircles divide AB¯\overline{AB} into four line segments of equal length, as shown. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicircles? Express your answer in terms of π\pi and in simplest radical form. [asy] fill((0,2)..(2,0)--(-2,0)..cycle,gray(0.7)); fill((-1,1)..(0,0)--(-2,0)..cycle,white); fill((1,1)..(0,0)--(2,0)..cycle,white); fill((0,1)..(1,0)--(-1,0)..cycle,white); draw((0,1)..(1,0)--(-1,0)..cycle,dashed); draw((0,2)..(2,0)--(-2,0)..cycle); label("AA",(-2,0),W); label("BB",(2,0),E); label("1",(-1.5,0),S); label("2",(0,0),S); label("1",(1.5,0),S); dot((0,0)); dot((-1,0)); dot((1,0)); draw((-2,-0.1)--(-2,-0.4)); draw((-1,-0.1)--(-1,-0.4)); draw((2,-0.1)--(2,-0.4)); draw((1,-0.1)--(1,-0.4)); [/asy]
Plain-text mathematical notation (without MathML)
Three semicircles of radius 1 are constructed on diameter (AB)¯ of a semicircle of radius 2. The centers of the small semicircles divide (AB)¯  into four line segments of equal length, as shown. What is the area of the shaded region that lies  within the large semicircle but outside the smaller semicircles? Express your answer in terms of π and in simplest radical form.

[asy]
fill((0,2)..(2,0)--(-2,0)..cycle,gray(0.7));
fill((-1,1)..(0,0)--(-2,0)..cycle,white);
fill((1,1)..(0,0)--(2,0)..cycle,white);
fill((0,1)..(1,0)--(-1,0)..cycle,white);
draw((0,1)..(1,0)--(-1,0)..cycle,dashed);
draw((0,2)..(2,0)--(-2,0)..cycle);
label("A",(-2,0),W);
label("B",(2,0),E);
label("1",(-1.5,0),S);
label("2",(0,0),S);
label("1",(1.5,0),S);
dot((0,0));
dot((-1,0));
dot((1,0));
draw((-2,-0.1)--(-2,-0.4));
draw((-1,-0.1)--(-1,-0.4));
draw((2,-0.1)--(2,-0.4));
draw((1,-0.1)--(1,-0.4));
[/asy]
Original LaTeX notation
Three semicircles of radius 1 are constructed on diameter $\overline{AB}$ of a semicircle of radius 2. The centers of the small semicircles divide $\overline{AB}$  into four line segments of equal length, as shown. What is the area of the shaded region that lies  within the large semicircle but outside the smaller semicircles? Express your answer in terms of $\pi$ and in simplest radical form.

[asy]
fill((0,2)..(2,0)--(-2,0)..cycle,gray(0.7));
fill((-1,1)..(0,0)--(-2,0)..cycle,white);
fill((1,1)..(0,0)--(2,0)..cycle,white);
fill((0,1)..(1,0)--(-1,0)..cycle,white);
draw((0,1)..(1,0)--(-1,0)..cycle,dashed);
draw((0,2)..(2,0)--(-2,0)..cycle);
label("$A$",(-2,0),W);
label("$B$",(2,0),E);
label("1",(-1.5,0),S);
label("2",(0,0),S);
label("1",(1.5,0),S);
dot((0,0));
dot((-1,0));
dot((1,0));
draw((-2,-0.1)--(-2,-0.4));
draw((-1,-0.1)--(-1,-0.4));
draw((2,-0.1)--(2,-0.4));
draw((1,-0.1)--(1,-0.4));
[/asy]

level

Level 5

type

Geometry

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Source and history

task_id
ee379031-8feb-534d-80f7-f1c3ff8e78e0
task_key
geometry--test--ee379031-8feb-534d-80f7-f1c3ff8e78e0
task_revision_id
1
config / split
geometry / test

Official source

initial import