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MATH / A quarter-circle of radius 3 units is drawn at each of the vertices of a square…

Problem

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

problem

A quarter-circle of radius 3 units is drawn at each of the vertices of a square with sides of 6 units. [asy] draw((0,0)--(0,1)--(1,1)--(1,0),black); fill((0,0)--(0,1)--(1,1)--(1,0)--cycle,black); fill((0.5,1){down}..{left}(0,0.5)--(0,1)--(0.5,1)--cycle,white); fill((0.5,1){down}..{right}(1,0.5)--(1,1)--(0.5,1)--cycle,white); fill((0.5,0){up}..{left}(0,0.5)--(0,0)--(0.5,0)--cycle,white); fill((0.5,0){up}..{right}(1,0.5)--(1,0)--(0.5,0)--cycle,white); [/asy] The area of the shaded region can be expressed in the form a−bπa-b\pi square units, where aa and bb are both integers. What is the value of $a+b?$
Plain-text mathematical notation (without MathML)
A quarter-circle of radius 3 units is drawn at each of the vertices of a square with sides of 6 units. [asy]

draw((0,0)--(0,1)--(1,1)--(1,0),black);

fill((0,0)--(0,1)--(1,1)--(1,0)--cycle,black);

fill((0.5,1){down}..{left}(0,0.5)--(0,1)--(0.5,1)--cycle,white);

fill((0.5,1){down}..{right}(1,0.5)--(1,1)--(0.5,1)--cycle,white);

fill((0.5,0){up}..{left}(0,0.5)--(0,0)--(0.5,0)--cycle,white);

fill((0.5,0){up}..{right}(1,0.5)--(1,0)--(0.5,0)--cycle,white);

[/asy] The area of the shaded region can be expressed in the form a−bπ square units, where a and b are both integers. What is the value of $a+b?$
Original LaTeX notation
A quarter-circle of radius 3 units is drawn at each of the vertices of a square with sides of 6 units. [asy]

draw((0,0)--(0,1)--(1,1)--(1,0),black);

fill((0,0)--(0,1)--(1,1)--(1,0)--cycle,black);

fill((0.5,1){down}..{left}(0,0.5)--(0,1)--(0.5,1)--cycle,white);

fill((0.5,1){down}..{right}(1,0.5)--(1,1)--(0.5,1)--cycle,white);

fill((0.5,0){up}..{left}(0,0.5)--(0,0)--(0.5,0)--cycle,white);

fill((0.5,0){up}..{right}(1,0.5)--(1,0)--(0.5,0)--cycle,white);

[/asy] The area of the shaded region can be expressed in the form $a-b\pi$ square units, where $a$ and $b$ are both integers. What is the value of $a+b?$

level

Level 2

type

Geometry

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Source and history

task_id
cc4a80e6-d8cc-5977-96b4-15b33e8c43ea
task_key
geometry--test--cc4a80e6-d8cc-5977-96b4-15b33e8c43ea
task_revision_id
1
config / split
geometry / test

Official source

initial import