Basic Geometry and Mensuration Notes for UPSC CSAT with Solved

Geometry and mensuration are a part of the basic numeracy section of the UPSC CSAT paper. The questions are at Class X level, and they test your knowledge of lines, angles, triangles, circles and the area and volume of common shapes. Since CSAT is a qualifying paper that needs 33 per cent marks, a few direct questions from this area can help you cross the line. This note explains the basic geometry facts, the formulas of area and volume, solved examples and the time-saving tricks, in simple language for students of any background.

Lines and Angles

An angle is the opening between two lines that meet at a point, measured in degrees. An acute angle is less than 90 degrees, a right angle is exactly 90 degrees, an obtuse angle is between 90 and 180 degrees, a straight angle is 180 degrees and a reflex angle is more than 180 degrees. Two angles are complementary if they add up to 90 degrees and supplementary if they add up to 180 degrees. When two lines intersect, the vertically opposite angles are equal. When a line crosses two parallel lines, the corresponding angles and the alternate angles are equal, and the co-interior angles add up to 180 degrees.

Triangles

• The sum of the three angles of a triangle is always 180 degrees, and an exterior angle equals the sum of the two opposite interior angles.
• By sides, a triangle is equilateral (all sides equal, each angle 60 degrees), isosceles (two sides equal) or scalene (no sides equal). By angles, it is acute, right-angled or obtuse.
• The sum of any two sides is greater than the third side.
• Pythagoras theorem: In a right-angled triangle, (hypotenuse) squared = (base) squared + (height) squared. Common triples: 3-4-5, 5-12-13, 8-15-17 and 7-24-25, and their multiples.
• Similar triangles have equal angles and proportional sides; the ratio of their areas is the square of the ratio of their corresponding sides.

Polygons and Circles

The sum of the interior angles of a polygon with n sides is (n - 2) x 180 degrees, so it is 360 degrees for a quadrilateral, 540 for a pentagon and 720 for a hexagon. The sum of the exterior angles of any polygon is 360 degrees. In a circle, the angle in a semicircle is 90 degrees, the angle at the centre is twice the angle at the circumference on the same arc, the tangent is perpendicular to the radius at the point of contact, and the two tangents from an outside point are equal in length.

Area and Perimeter of Plane Figures

Use pi = 22/7 when the radius is a multiple of 7, and 3.14 otherwise. For a triangle with sides a, b and c, Heron's formula gives the area as the square root of s(s - a)(s - b)(s - c), where s is half the perimeter.

Surface Area and Volume of Solids

Remember the unit conversions: 1 cubic metre equals 1,000 litres, and 1 litre equals 1,000 cubic centimetres. When a solid is melted and recast into another shape, the volume remains the same.

Solved Examples

1. The two sides of a right triangle are 6 cm and 8 cm. Find the hypotenuse. It is the square root of (36 + 64) = 10 cm.
2. Find the area of an equilateral triangle of side 10 cm. Area = (root 3 / 4) x 100 = 25 root 3, about 43.3 square cm.
3. Find the circumference and the area of a circle of radius 7 cm. Circumference = 2 x 22/7 x 7 = 44 cm; area = 22/7 x 49 = 154 square cm.
4. Find the volume and the total surface area of a cuboid of 5 cm x 4 cm x 3 cm. Volume = 60 cubic cm; surface area = 2(20 + 12 + 15) = 94 square cm.
5. Find the volume of a cylinder of radius 7 cm and height 10 cm. Volume = 22/7 x 49 x 10 = 1,540 cubic cm.
6. Find the slant height and the volume of a cone with radius 3 cm and height 4 cm. Slant height = root of (9 + 16) = 5 cm; volume = 1/3 x pi x 9 x 4 = 12 pi, about 37.7 cubic cm.
7. Find the sum of the interior angles of a hexagon. (6 - 2) x 180 = 720 degrees.
8. The diagonal of a square is 10 root 2 cm. Find its area. Side = diagonal / root 2 = 10 cm, so the area is 100 square cm.

Shortcuts and Tips for CSAT

• If the radius of a circle is doubled, its area becomes four times; if the side of a square is doubled, the area becomes four times, and the volume of a cube becomes eight times.
• For a rectangle with a fixed perimeter, the square has the maximum area.
• When the length is increased by x per cent and the breadth is decreased by y per cent, the change in the area is better done by taking the base value of 100 for each side.
• Learn the Pythagorean triples, as they save time in the right triangle questions.
• Draw a quick rough figure for every question and mark the given values.

Common Mistakes

Students often mix up the radius and the diameter, forget to square the radius in the area formula, or use the slant height in the place of the vertical height in the volume of the cone. Another common error is not converting the units, such as centimetres and metres, before calculation. Read the question carefully for whether the "curved" or the "total" surface area is required. Since CSAT is not a test of heavy calculation, always check the options first: often a rough estimate with pi as 3 can remove the wrong options.

Practice Questions With Answers

• Q1. Find the area of a circle of diameter 14 cm. Answer: 154 square cm.
• Q2. The sides of a triangle are 5, 12 and 13. Is it a right triangle? Answer: Yes, since 25 + 144 = 169.
• Q3. Find the volume of a cube of side 6 cm. Answer: 216 cubic cm.
• Q4. If two angles of a triangle are 50 and 60 degrees, find the third. Answer: 70 degrees.
• Q5. Find the perimeter of a rectangle of 12 cm by 8 cm. Answer: 40 cm.

Revise the number-based topics with our number system notes and the CSAT strategy guide.

How to Practise This Topic

Spend one hour on formulas and one hour on practice every day for four days. On the first day, learn the angle and triangle facts with ten questions. On the second day, learn the area formulas with ten questions. On the third day, learn the solids with ten questions. On the fourth day, take a mixed test of twenty questions in thirty minutes and note your mistakes. Make a one-page formula chart and paste it on your study table, so that you read it daily.

Quick Revision

• Triangle angles sum to 180 degrees; Pythagoras theorem; common triples 3-4-5 and 5-12-13.
• Circle: area pi r squared; circumference 2 pi r.
• Cuboid: volume lbh; cylinder: pi r squared h; cone: 1/3 pi r squared h; sphere: 4/3 pi r cubed.
• Polygon interior angles: (n - 2) x 180 degrees.

Frequently Asked Questions (FAQ)

What is the sum of the angles of a triangle?

The sum of the three angles of a triangle is 180 degrees.

What is the Pythagoras theorem?

In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

What are common Pythagorean triples?

3-4-5, 5-12-13, 8-15-17 and 7-24-25 are common triples.

What is the area of a circle?

The area of a circle is pi times the radius squared.

What is the circumference of a circle?

The circumference is 2 pi times the radius.

What is the volume of a cylinder?

The volume of a cylinder is pi times the radius squared times the height.

What is the volume of a cone?

The volume of a cone is one-third of pi times the radius squared times the height.

What is the volume of a sphere?

The volume of a sphere is four-thirds of pi times the radius cubed.

How do I find the sum of the interior angles of a polygon?

Use (n - 2) times 180 degrees, where n is the number of sides.

How many litres are there in 1 cubic metre?

One cubic metre equals 1,000 litres.

What is the area of an equilateral triangle?

The area is root 3 divided by 4 times the side squared.

Is geometry important for CSAT?

Yes. A few direct questions on geometry and mensuration are common in the basic numeracy section, and they help in crossing the qualifying marks.