Geometry

UPSC CSAT Maths

Table of contents

Basic conversions

A. 1 m = 100 cm = 1000 mm, 1 km = 1000 m
B. 1 inch = 2.54 cm
C. 100 kg = 1 quintal
D. 10 quintal = 1 tonne = 1000 kg
E. 1 litre = 1000 cc, 1000 lit = 1 cubic metre
F. 1 acre = 100 sq m


Angles

  • Acute angle: An angle whose measure is less than 90 degrees. T
  • Right angle: An angle whose measure is 90 degrees.
  • Obtuse Angle: An angle whose measure is bigger than 90 degrees but less than 180 degrees.
  • Straight angle: Is an angle whose measure is 180 degrees
  • Reflex angle: An angle whose measure is more than 180 degrees but less than 360 degrees.
  • Full Rotation Angle: An angle whose measure is 360 degrees
  • Complementary angles: Two angles whose measures add to 90 degrees
  • Supplementary Angles: Two angles whose measures add up to 180 degrees.

Triangles

  • A triangle is a polygon having three sides. Sum of all the angles of a triangle = 180°.
  • Properties (General)
    • Sum of the length of any two sides of a triangle has to be always greater than the third side.
    • Side opposite to the greatest angle will be the greatest and the side opposite to the smallest angle the Smallest.
  • Area:
    • Area = 1/2 base * height  (A=1/2 * bh)
Height = Perpendicular distance between the base and vertex opposite to it
    • Area = s(s - a)(s - b)(s - c) (Heron’s formula) 
where s = (a+b+c)/2 (a, b and c being the length of the sides)
  • Congruence of triangles: Two triangles are congruent if all the sides of one are equal to the corresponding sides of another.
  • Two triangles are said to be similar when they have two corresponding angles congruent and the sides proportional. The area, altitude, and volume of Similar triangles are in the same ratio as the ratio of the length of their sides.

Circles


Practice Questions

1. A rectangular plot of lawn shown in the figure has dimensions 'x' and 'y' and is surrounded by a gravel pathway of width 2 m. What is the total area of the Gravel Pathway?
a. 2x + 2y + 4  

b. 2x + 2y + 8  

c. 4x + 4y + 8    

d. 4x + 4y + 16

2. Find the perimeter of the rectangular plot of land whose area is 660 m 2 and whose length is 33 m.
a. 200m

b. 106m  

c. 53m  

d. 100m

3. The length and breadth of a rectangular courtyard is 75 m and 32 m. Find the cost of levelling it at the rate of ₹3 per m2. Also, find the distance covered by a boy to take 4 rounds of the courtyard.
a. ₹7200 & 428
b. ₹6200 & 428
c. ₹6200 & 256
d. ₹7200 & 428

4. The breadth of the rectangle is 8 cm and its diagonal is 17 cm. Find the area of the rectangle.
a. 150cm2
b. 120 cm2
c. 100 cm2
d. 200 cm2

5. Length of a side of a square field is 275 m. What will be the cost of levelling the field at a rate of 10 cent per square metre?
a. 756250 cents
b. 758250 cents
c. 766250 cents
d. 856250 cents

6. An error 2% in excess is made while measuring the side of a square. The percentage of error in the calculated area of the square is:
a. 3.96 %
b. 4.04%
c. 4%
d. 8%

7. Find the area of the triangle whose sides are 15cm, 20cm and 25cm respectively.
a. 100cm2
b. 180cm2
c. 150cm2
d. 250cm2

8. Find the area and perimeter of a right-angled triangle with base 6 cm and hypotenuses is 6.5cm.
a. 12cm
d. 20cm
c. 10cm
d. 15cm

9. If the radius of a circle is decreased by 10%, then the total decrease in the area of the circle is.
a. 20%
b. 16%
c. 19%
d. 20.5%

10. If in a triangle area is numerically equal to the perimeter, then the radius of the inscribed circle of the triangle is
a. 1
b. 1.5
c. 2
d. None of these

11. The area of equilateral triangle is 900√3 sq. m. What is its perimeter?
a. 120m
b. 150m
c. 180m
d. 135m

12. If the diagonal of square is 4√2, then its perimeter is
a. 8
b. 15
c. 10
d. 16

Answer Key

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