These geometry challenges are designed to push your problem-solving skills and deepen your understanding of shapes, angles, and spatial reasoning. Each question encourages critical thinking and includes step-by-step solutions so you can check your reasoning or learn a new strategy.
Challenge Problems
\(\textbf{1)}\) Find the area of this rectangle.
The area is \( 8+4\sqrt{3}\) cm\(^2 \)
\(\textbf{2)}\) Find the area of the red square.
The area is \( 3-2\sqrt{2}\approx 0.17157\) units\(^2 \)
\(\textbf{3)}\) Find the radius of the green circle.
The radius is \(8\) units
\(\textbf{4)}\) Find the area of the blue region.
The area is \(5\) cm\(^2\)
\(\textbf{5)}\) Find the area of the pink region.
The area is \(7\) cm\(^2\)
\(\textbf{6)}\) Find the perimeter.
The perimeter is \(68\) units
\(\textbf{7)}\) Solve for x.
\(x=30\)
\(\textbf{8)}\) Show that the Area \(=2x^2+24x+46\).
\(\textbf{9)}\) What is the area of the blue region?
The area is \(1296+81\pi\) units\(^2\)
\(\textbf{10)}\) Find the area of the green triangle.
The area is \(\displaystyle\frac{450\sqrt{3}}{2}\) units\(^2\)
\(\textbf{11)}\) Find the area of the blue region.
The area is \(25-431.25\pi+300\sqrt{2}\pi\) cm\(^2\)
\(\textbf{12)}\) Find the measure of the angle “?”.
The angle is \(51.05^{\circ}\)
\(\textbf{13)}\) Find the area of the red region.
The area is \(84\) units\(^2\)
\(\textbf{14)}\) Find the area of the large circle.
The area is \(420\) m\(^2\)
\(\textbf{15)}\) How many degrees is the red angle?
The red angle is \(135^{\circ}\)
\(\textbf{16)}\) Find the area of the grey rectangle.
The area is \(42\) cm\(^2\)
\(\textbf{17)}\) Solve for x.
The answer is \(x=11\)
\(\textbf{18)}\) Find the area of the blue rectangle.
The area is \(25\) cm\(^2\)
\(\textbf{19)}\) Red line is tangent to blue circle. Find the value of r.
r is \(\displaystyle\frac{12}{5}\) units
\(\textbf{20)}\) Given 2 squares and 2 triangles, find the area of the red triangle.
Area is \(\displaystyle\frac{9\sqrt{3}}{2} \approx 7.79\) square units
\(\textbf{21)}\) Find the area of the red region.
Area is \(75-\frac{25 \pi}{2} \approx 35.73\) square units
\(\textbf{22)}\) Find the area of the orange triangle.
Area is \(3\sqrt{15} \approx 11.62\) square units
\(\textbf{23)}\) Blue Area = Red Area = Green Area. Find the Area of the Square.
The area is \(117\) cm\(^2\)
\(\textbf{24)}\) Find the Area of the Red Square
The area is \(4\) m\(^2\)
\(\textbf{25)}\) Find the Area of the Blue Rectangle
The area is \(20\) square units\(\)
\(\textbf{26)}\) Find the Area of the Red Region
The area is \(\displaystyle 6-9\pi\frac{\sin^{-1}{\left(\frac{3}{5}\right)}}{360}-4\pi\frac{\sin^{-1}{\left(\frac{4}{5}\right)}}{360}-\frac{\pi}{4}\) m\(^2\)
\(\approx 0.464256\) m\(^2\)
\(\approx 0.464256\) m\(^2\)
\(\textbf{27)}\) Find the ratio of \(\displaystyle\frac{\text{Pink Area}}{\text{Green Area}}\)
The answer is \(\displaystyle\frac{4\pi+2\pi\sqrt{2}-4}{4-\pi} \approx 20.3308\)
\(\textbf{28)}\) Find the area of the red region
The area is \(32\pi-64 \approx 36.53\)
\(\textbf{29)}\) Find the area of the red inscribed circle
The area is \(\pi \approx 3.14159\)
\(\textbf{30)}\) Find the radius of the circle
The radius is \(\displaystyle\frac{2\sqrt{2}}{1+\sqrt{2}} \approx 1.17157\)
\(\textbf{31)}\) Find the area of the square
The area is \(144\) square units
\(\textbf{32)}\) Given: 3 Squares and a Semicircle. What’s the total area of the 3 squares?
The area is \(14\) square units
\(\textbf{33)}\) Find the value of x.
\(x=30\)°
\(\textbf{34)}\) Find the area of the blue triangle.
\(8 \) cm\(^2 \)
\(\textbf{35)}\) Two equilateral Triangles. What’s the Angle?
Angle= \(30\)°
\(\textbf{36)}\) Solve for x, one of the sides of 3 given squares.
\(x=\displaystyle\frac{9-2\sqrt{3}}{46}\approx 0.120345617\)
\(\textbf{37)}\) Four Squares. What is the shaded area?
Area=\(36 \text{ } units^2\)
\(\textbf{38)}\) Two quarter circles and a semicircle. What is the shaded area?
Area=\(18 \pi \text{ } units^2\)
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