# CAT Practice : Mensuration

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The following topics are covered in the CAT quant section from Geometry - Mensuration. Detailed explanatory answers, solution videos and slide decks are also provided.
1. ### Cone and Frustum

A right circular cone has height H and radius R. A small cone is cut off at the top by a plane parallel to the base. At what height above the base the section has been made?

Statement (I): H = 20 cm
Statement (II): Volume of small cone: volume of large cone : 1:15

1. If the question can be answered with statement I alone but not statement II alone, or can be answered with statement II alone but not statement I alone.
2. If the question cannot be answered with statement I alone or with statement II alone, but can be answered if both statements are used together.
3. If the question can be answered with either statement alone.
4. If the question cannot be answered with the information provided.
• Correct AnswerChoice (B).
If the question cannot be answered with statement I alone or with statement II alone, but can be answered if both statements are used together.
• Data Sufficiency
• Hard
2. ### Spheres

A sphere of radius r is cut by a plane at a distance of h from its center, thereby breaking this sphere into two different pieces. The cumulative surface area of these two pieces is 25% more than that of the sphere. Find h.

1. ${r \over \sqrt{2}}$
2. ${r \over \sqrt{3}}$
3. ${r \over \sqrt{5}}$
4. ${r \over \sqrt{6}}$
• Correct AnswerChoice (A).
${r \over \sqrt{2}}$
• Sphere
• Hard
3. ### Circles and Chords

Two mutually perpendicular chords AB and CD meet at a point P inside the circle such that AP = 6 cms, PB = 4 units and DP = 3 units. What is the area of the circle?

1. ${125{\pi} \over 4}$ sq cms
2. ${100{\pi} \over 7}$ sq cms
3. ${125{\pi} \over 8}$ sq cms
4. ${52{\pi} \over 3}$ sq cms
4. ### Max No. of Cans

Cylindrical cans of cricket balls are to be packed in a box. Each can has a radius of 7 cm and height of 30 cm. Dimension of the box is l = 76 cm, b = 46 cm, h = 45 cm. What is the maximum number of cans that can fit in the box?

1. 15
2. 17
3. 22
4. 21
5. ### Area of Triangle

PQRS is a square of sides 2 cm & ST = 2 cm. Also, PT=RT. What is the area of ∆PST?

1. 2 cm2
2. √3 cm2
3. √2 cm2
4. ${\frac {1}{√2}}$ cm2
6. ### Length of String

A string is wound around two circular disk as shown. If the radius of the two disk are 40 cm and 30 cm respectively. What is the total length of the string?

1. 70 cm
2. 70 + 165*${\pi}$
3. 70 + 120${\pi}$
4. 70 + 165*${\frac {\pi}{2}}$
7. ### Cuboid

Figure above shows a box which has to be completely wrapped with paper. However, a single Sheet of paper need to be used without any tearing. The dimension of the required paper could be

1. 17 cm by 4 cm
2. 12 cm by 6 cm
3. 15 cm by 4 cm
4. 13 cm by 4 cm
8. ### Right Circular Cone

An inverted right circular cone has a radius of 9 cm. This cone is partly filled with oil which is dipping from a hole in the tip at a rate of 1cm3/hour. Currently the level of oil 3 cm from top and surface area is 36π cm2. How long will it take the cone to be completely empty?

1. 216π hours
2. 1 hour
3. 3 hours
4. 36π hours
9. ### Square and Triangle

A square PQRS has an equilateral triangle PTO inscribed as shown:

What is the ratio of A∆PQT to A∆TRU?

1. 1 : 3
2. 1 : √3
3. 1 : √2
4. 1 : 2
10. ### Spheres and Cubes

A spherical shaped sweet is placed inside a cube of side 5 cm such that the sweet just fits the cube. A fly is sitting on one of the vertices of the cube. What is the shortest distance the fly must travel to reach the sweet?

1. 2.5 cm
2. 5(√3 – 1) cm
3. 5(√2 – 1) cm
4. 2.5(√3 – 1) cm
11. ### Tomatoes in Backyard

Anil grows tomatoes in his backyard which is in the shape of a square. Each tomato takes 1 cm2 in his backyard. This year, he has been able to grow 131 more tomatoes than last year. The shape of the backyard remained a square. How many tomatoes did Anil produce this year?

1. 4225
2. 4096
3. 4356
4. Insufficient Data
12. ### Circles

PQRS is a circle and circles are drawn with PO, QO, RO and SO as diameters areas A and B are shaded A/B is equal to

1. ${\pi}$
2. 1
3. ${\frac{\pi}{4}}$
4. 2
13. ### Squares

ABCD is a square drawn inside a square PQRS of sides 4 cm by joining midpoints of the sides PQ, QR, RS, SP. Another square is drawn inside ABCD similarly. This process is repeated infinite number of times. Find the sum of all the squares.

1. 16 cm2
2. 28 cm2
3. 32 cm2
4. Infinite
14. ### Pentagon and Circles

PQRST is a pentagon in which all the interior angles are unequal. A circle of radius ‘r’ is inscribed in each of the vertices. Find the area of portion of circles falling inside the pentagon.

1. ${\pi}$r2
2. 1.5${\pi}$r2
3. 2${\pi}$r2
4. 1.25${\pi}$r2
15. ### Three Circles

Three circles with radius 2 cm touch each other as shown :-

Find the area of the circle, circumscribing the above figure.

1. 3${\pi}$(4+√3)2
2. ${\frac{\pi}{2}}$(4+2√3)2
3. ${\frac{\pi}{4}}$(4+2√3)2
4. ${12\pi - \frac{\pi}{2}}$(4+2√3)2
16. ### Concentric circles

There are 5 concentric circles that are spaced equally from each other by 1.25 cms. The innermost circle has a square of side √(32) cm inscribed in it. If a square needs to be inscribed in the outermost circle, what will be its area?

1. 324 sq. cm.
2. (66 + 40√2) sq. cm.
3. 210.125 sq. cm.
4. 162 sq.cm.
17. ### Cross section of sphere

A spherical rubber ball of radius 14 cm is cut by a knife at a distance of “x” cm from its centre, into 2 different pieces. What should be the value of “x” such that the cumulative surface area of the newly formed pieces is 3/28 more than the rubber ball’s original surface area?

1. 11.4 cm
2. 3 cm
3. 12.4 cm
4. 7.5 cm

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