Quantitative Aptitude

Geometry

1,950 Questions
All Circle Tangents and Chords 772 Coordinate Geometry 360 Circle and Arc Measures 216 Triangle Centers and Medians 175 Composite and Inscribed Shapes 174 Similarity, Congruence, and Proportionality 118 Right Triangles and Pythagorean Theorem 100 Standard 2D Mensuration 22 Polygon and Angle Sums 10 3D Solid Mensuration 3

Geometry Questions

Multiple choice

A right angled isosceles triangle is inscribed in a circle of radius r What is the area of the remaining portion of the circle?
  1. $\displaystyle \frac{\pi r^{2}}{2}$
  2. $\displaystyle \left ( \pi -\frac{1}{2} \right )r^{2}$
  3. $\left ( \pi -1 \right )r^{2}$
  4. $\left ( \pi -2 \right )r^{2}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

A right isosceles triangle inscribed in a circle has a hypotenuse equal to the diameter (2r). The legs are r * sqrt(2). The area of the triangle is (1/2) * base * height = (1/2) * (r * sqrt(2)) * (r * sqrt(2)) = r^2. The remaining area is the circle area minus triangle area: pi * r^2 - r^2 = (pi - 1) * r^2.

Multiple choice

What is the area of the region of the circle which is situated outside the inscribed square of side $x$?
  1. $\displaystyle \left ( \pi -2 \right )x^{2} $
  2. $\displaystyle \left ( \pi -2 \right )\frac{x^{2}}{2} $
  3. $\displaystyle 2\left ( \pi -2 \right )x^{2} $
  4. $\displaystyle \left ( \pi -2 \right )\frac{x^{2}}{4} $
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The square is inscribed in a circle, so the diagonal of the square equals the diameter of the circle. Diagonal = x*sqrt(2), so radius r = (x*sqrt(2))/2. Area of circle = pi * r^2 = pi * (x^2 / 2). Area of square = x^2. The difference is x^2(pi/2 - 1) = (pi - 2) * x^2 / 2.

Multiple choice

A chord AB of a circle of radius $10$cm makes a right angle at the centre of the circle. Find the area of the major and minor segments (Take $\pi=3.14$)
  1. Area of major segment$=285.5cm^2$,  Area of minor segment$=28.5cm^2$
  2. Area of major segment$=232cm^2$,  Area of minor segment$=24.6cm^2$
  3. Area of major segment$=212.7cm^2$,  Area of minor segment$=32.5cm^2$
  4. Area of major segment$=196.2cm^2$,  Area of minor segment$=21.3cm^2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For a circle with radius 10 and a 90-degree central angle, the area of the minor segment is (pi*r^2 * 90/360) - (0.5 * r^2 * sin(90)) = (3.14 * 100 / 4) - 50 = 78.5 - 50 = 28.5 cm^2. The major segment area is total area - minor segment = 314 - 28.5 = 285.5 cm^2.

Multiple choice

Find the area of a segment of a circle of radius $21\ cm$, if the angle made by the arc of the segment has a measure of $60^{\circ}$?
  1. $45.27\ cm^{2}$
  2. $40.27\ cm^{2}$
  3. $40.8\ cm^{2}$
  4. $44.27\ cm^{2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Area of segment = Area of sector - Area of triangle = (theta/360 * pi * r^2) - (1/2 * r^2 * sin(theta)). For theta=60, area = (60/360 * 3.14 * 21^2) - (sqrt(3)/4 * 21^2) = (1/6 * 3.14 * 441) - (0.433 * 441) = 230.86 - 190.95 = 39.91. Close to 40.27.