Circles Inscribed in Squares: 7 Hard Geometry Problems & Solutions
Geometry problems involving circles inscribed in squares and rectangles are common on standardized tests such as the GMAT, GED, and SAT. Typical problems ask you to solve for the radius or area of a figure, or find the ratio of measurements between various figures. Most problems can be solved by applying properties of circles and and the Pythagorean Theorem.
For these problems, the most important geometric property of circles is that if two circles are tangent to one another, then the line joining their centers also passes through the circles' point of tangency. Another property to remember is that any line segment that connects the center of a circle to a point on the circumference has a length equal to the radius.
Try your hand at these 7 classic problems plus 2 bonus problems.
A circle with a radius of 1 is inscribed within a 2-by-2 square. A smaller circle is inscribed in the corner such that it is tangent to two adjacent sides of the square and the larger circle. What is the radius of this smaller circle?
Solution: Call the radius of the smaller circle r. If you connect the centers of the two circles with a straight line, it will pass through the point of tangency and this line segment will have a length of 1 + r. This line makes a 45-degree angle with the corner of the square. As shown in the figure on the right, you can add two more lines to form an isosceles right triangle whose hypotenuse is 1 + r, and whose two legs are both 1 - r.
Since it is an isosceles right triangle, the hypotenuse must be sqrt(2) times longer than each leg. Thus we have the equation
1 + r = sqrt(2)(1 - r)
r + sqrt(2)r = sqrt(2) - 1
r(1 + sqrt(2)) = sqrt(2) - 1
r = [sqrt(2) -1]/[sqrt(2) + 1]
r = 3 - 2*sqrt(2)
r ≈ 0.171573
Two circles each with a radius of 1 and tangent to each other are inscribed in a 2-by-4 rectangle. A smaller circle is inscribed in the space between the circles and the long edge of the rectangle, such that it is tangent to both circles and the edge of the rectangle. What is the radius of this smaller circle?
Solution: A right triangle can be formed that connects the center of the smaller circle, the center of one of the larger circles, and the tangent point between the two larger circles. Call the radius of the smaller circle r. Then the legs of this triangle have lengths of 1 and 1 - r, and the hypotenuse has a length of 1 + r. This gives us the following Pythagorean equation to solve for r:
1^2 + (1-r)^2 = (1+r)^2
1 + 1 - 2r + r^2 = 1 + 2r + r^2
2 - 2r = 1 + 2r
1 = 4r
r = 1/4.
A circle is inscribed within a quarter-circle sector. It is tangent to the arc of the sector and the two perpendicular radii of the sector. What is the ratio of the area of the circle to the area of the sector?
Solution: Since this is a ratio problem, it does not matter what the radius of each circle is, only their sizes relative to one another. Therefore we can simplify the problem by setting the radius of the sector equal to 1. If we draw a line from the vertex of the sector to the point on the arc where it is tangent to the circle, this line segment has a length of 1 because it is a radius.
If we call the radius of the smaller circle r, we can see that the length of this line segment is also equal to r + sqrt(2)r. Therefore we can solve for r:
r + sqrt(2)r = 1
[1 + sqrt(2)]r = 1
r = 1/[1 + sqrt(2)]
r = sqrt(2) - 1
The area of the quarter circle is π/4. The area of the smaller inscribed circle is π(3 - 2*sqrt(2)). Thus, the ratio of the circle to the sector is
[ π(3 - 2*sqrt(2)) ] / [ π/4 ]
= 12 - 8*sqrt(2)
Four circles, each with a radius of 1, are arranged so that they are tangent to two others and their centers form the corners of a square. A smaller circle is inscribed in the space bounded by the four circles, and this smaller circle is tangent to each of the other four. What is its radius?
Solution: The square formed by the four circles has an edge length of 2. The diagonal of this square is 2*sqrt(2). If the radius of the smaller circle is r, the diagonal of the square is also equal to 1 + 2r + 1. Thus we can solve for r:
1 + 2r + 1 = 2*sqrt(2)
2 + 2r = 2*sqrt(2)
1 + r = sqrt(2)
r = sqrt(2) - 1
r ≈ 0.41421
A circle is inscribed within a square. A smaller square is inscribed within this circle. What is the ratio of the area of the larger square to that of the smaller square?
Solution: This is a classic math brain teaser question that requires no sophisticated geometry or algebra knowledge. Spin the circle so that the corners of the inscribed square lie on the centers of the edges of the larger square. Now you can see that the smaller square occupies exactly half the space of the larger square. Therefore the area ratio is 2:1.
A square is inscribed in a circle. Within this square is inscribed a smaller circle. What is the ratio of the area of the larger circle to that of the smaller circle?
Solution: Inscribe another smaller square within the smaller circle. Call the are of the large square Q, the area of the small square q, the area of the large circle M, and the area of the small circle m. We know that M/Q = m/q by properties of scale. And from the previous problem we know that Q/q = 2. Therefore
M/Q = m/q
Mq = mQ
M/m = Q/q
M/m = 2
So the ratio of the area of the large circle to the area of the small circle is also 2:1.
The figure above shows a square with a side length of 1. Along the edges are four half-circles with each a radius of 0.5. Their intersections inside the square create eight regions: four like the one shown in pink, and four like the one shown in yellow. What are the areas of the pink and yellow regions?
Solution: Call the area of the pink region P and the area of the yellow region Y. We know that 4P + 4Y equals the area of the square. We also know that 2Y + P equals the area of one half-circle within the square. This information gives us a set of two equations in two unknowns.
4P + 4Y = 1
2Y + P = π/8
If we multiply the second equation by 2 and subtract it from the first we get
2P = 1 - π/4
P = 1/2 - π/8
P ≈ 0.1073
Solving back for Y gives us
Y = π/8 - 1/4
Y ≈ 0.1427
BONUS Problem #1
Two circles of equal size are inscribed in a square such that the circles are tangent to each other and to two sides of the square. The centers of the circle lie on the square's diagonal. If the radius of each circle is 1, what is the side length of the square?
Solution: As shown in the figure at left, if you connect the centers of the squares with a line segment, this line segment can be made into the hypotenuse of an isosceles right triangle. Because this line segment passes through the point of tangency between the two circles, its length is 2. The other two sides of the triangle have a length of 2/sqrt(2), which is equivalent to sqrt(2).
Now draw a line segment that connects the center of a circle with the point on the square's edge where it is tangent. Do this for both circles for all four points of tangency on the square's boundary. You can now see that the side length of the square is equal to
1 + sqrt(2) + 1
= 2 + sqrt(2).
BONUS Problem #2
Three circles have a radius of 6 and are inscribed in a rectangle as shown in the figure above. What is the area and perimeter of the rectangle?
Solution: It's easy to see that the width of the rectangle is four times the radius of the circles, 4*6 = 24. The height of the rectangle is a little harder to work out, but it can be figured in the same way as in Bonus Problem #1 by making a triangle with the radii and centers of the circles, as in the image on the right.
The triangle formed by the centers of the circles is equilateral with a side length of 12, and its height is 6*sqrt(3). Therefore, the height of the bounding rectangle is 6 + 6*sqrt(3) + 6 = 12 + 6*sqrt(3).
The area of the rectangle is 24*(12 + 6*sqrt(3)) = 288 + 144*sqrt(3). The perimeter of the rectangle is 24 + 12 + 6*sqrt(3) + 24 + 12 + 6*sqrt(3) = 72 + 12*sqrt(3).
Some Related Geometry Facts and Formulas
- If a square and a circle have equal area, then the diameter of the circle is 2/sqrt(π) times the side length of the square. Equivalently, the side length of the square is sqrt(π)/2 times the diameter of the circle.
- If a square and circle have equal perimeter, then the diameter of the circle is 4/π times the side length of the square. Equivalently, the side length of the square is π/4 times the diameter of the circle.
- In terms of area, the largest quadrilateral that can be inscribed in a circle is a square.
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