A cylindrical vessel is filled with water up to some height. When a sphere of diameter 8 cm is dropped into the cylinder the water level rises by half of the initial level. When a sphere of diameter 16 cm is dropped the water level rises by some other value. What percentage of this new height is the initial level of water? Answer: 20% Explanation: Let the radius of the … [Read more...] about Solid Geometry – Sphere and Cylinder Volumes

# GRE Geometry, Solid Geometry, Coordinate Geometry

## Solid Geometry Mensuration – Circles

Chord AC at a distance of 7 cm from the center of a circle subtends an angle of 120 degrees at the center. Find the area of major segment. Answer: 1232/3 + 49*root(3) Explanation: In triangle OCD, Angle (COD) = 60 degrees Angle (OCD) = 30 degrees Hence tan(OCD) = OD/CD CD = 7*root(3) Sin(OCD) = OD/OC OC = 14 Hence AC = 14*root(3) Area of triangle AOC = … [Read more...] about Solid Geometry Mensuration – Circles

## Geometry Triangle – Equating Areas

In a triangle ABC, AB = 16, AC = 9. Find the length of the altitude to BC if the circumradius is 9. Answer: 8 Explanation: Consider the figure below. We know that, Area of a triangle = abc/4R where a,b,c are the sides of the triangle and R is the circumradius. Hence, (AB x BC x AC)/4R = (1/2) x BC x AD (16 x BC x 9)/36 = (1/2) x BC x AD Therefore, AD = 8. … [Read more...] about Geometry Triangle – Equating Areas

## Geometry – Obtuse Angled Triangles

How many obtuse-angled triangles can you form with 10,24,x as the sides where x is an integer? Answer: 14 Explanation: For three numbers to form an obtuse-angled triangle the square of the largest number should be greater than the sum of the squares of the other two numbers. For three numbers to form a triangle the sum of any two numbers should be greater than the … [Read more...] about Geometry – Obtuse Angled Triangles

## Geometry Circles – Angle in Same Segment

Q,R,S,T are four points on the circumference of a circle. QT is the perpendicular bisector of RS. PQ is a tangent drawn to the circle from an external point P. If angle(SQP) = 20, find angle(TRS). Answer: 70 degrees Explanation: Consider the diagram below. QT is the perpendicular bisector of a chord RS. Hence QT has to pass through the centre of the … [Read more...] about Geometry Circles – Angle in Same Segment