Chapter 4 of 5

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CUET UG Quantitative & Numerical Ability★ ~2-4 of the General Test's 50 questions

Algebra, Geometry & Mensuration

Basic algebra (equations, substitution), basic geometry (angle rules for triangles and straight lines), and mensuration (perimeter, area and volume of the standard shapes) — three Class 8 topics tested together in one compact chapter.

⏱ 8 min readEasy difficulty✓ Free · exam-pattern aligned

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Timed practiceMCQs from this chapter with instant scoring

FlashcardsFormulas & traps on spaced repetition

CUET UG PYQsSolved previous-year questions

By the end of this chapter you'll be able to…

1Solve a linear equation with the variable appearing on both sides
2Evaluate an algebraic expression by substitution followed by BODMAS
3Apply the fixed angle-sum facts for a straight line (180°), a point (360°) and a triangle (180°)
4Solve for a triangle's angles when given as a ratio
5Apply the perimeter and area formulas for rectangles, squares and triangles, including right triangles
6Apply the circumference and area formulas for a circle
7Apply the volume formulas for a cube and a cylinder

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Why this chapter matters in CUET UG

This chapter bundles three short, separate Class 8 topics — solving basic equations, applying fixed angle rules, and computing perimeter/area/volume with standard formulas. Each sub-topic contributes only a question or two, but all three are pure formula-application: there is no shortcut around knowing the correct fixed rule.

Before you start — revise these

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Numerical Ability

Solving a simple one-sided linear equation, taught there, is extended here to equations with the variable on both sides.

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Ratio simplification and ratio sharing

Angles given as a ratio are solved with the identical 'find one part, then scale up' method used for ratio-sharing.

Algebra, Geometry & Mensuration — CUET UG General Test

This chapter combines three genuinely separate Class 8 topics that the General Test tests together in small numbers each: basic algebra (equations and substitution), basic geometry (the fixed angle rules every triangle and straight line obeys), and mensuration (the standard perimeter, area and volume formulas). Each sub-topic is short, but all three demand the same discipline — apply the correct fixed rule or formula without skipping a step.

1. Linear equations with the variable on both sides

When a variable appears on both sides of an equation, collect all variable terms onto one side and all constant terms onto the other, changing sign for any term moved across the equals sign — exactly the same move as a simpler one-sided equation, just applied twice. $4 x - 5 = 2 x + 11$ becomes $4 x - 2 x = 11 + 5$ , giving $2 x = 16$ and $x = 8$ .

2. Evaluating algebraic expressions

Evaluating an expression means substituting given numeric values for its variables and then following BODMAS exactly, treating the substituted numbers no differently from any other arithmetic expression. Given $x = 4$ and $y = 3$ , the expression $2 x^{2} - 3 y + 5$ evaluates as $2 (16) - 3 (3) + 5 = 32 - 9 + 5 = 28$ — powers are resolved before multiplication, which happens before addition and subtraction.

3. Angles on a straight line and at a point

Angles that together form a straight line always sum to exactly 180°, and angles that together form a full turn around a single point always sum to exactly 360° — both are fixed facts, never something to calculate from other information. If three angles on a straight line are given as $2 x$ , $3 x$ and $x$ , their sum must equal 180°, giving $6 x = 180$ , so $x = 30$ and the three angles are 60°, 90° and 30°.

4. Triangle angle sum and angles in a ratio

Every triangle's three interior angles sum to exactly 180°, regardless of the triangle's shape or size — this single fact is enough to find a missing angle whenever the other two are known. If two angles of a triangle are 50° and 65°, the third is $180 - 50 - 65 = 65°$ .

When a triangle's angles are given as a ratio rather than individual values, the ratio's terms are treated as "parts" of the 180° total — find one part's value first, then scale each angle up, exactly as in ratio-sharing. A triangle with angles in the ratio 2:3:4 splits 180° into 9 parts of 20° each, giving angles of 40°, 60° and 80°.

5. Perimeter and area of standard shapes

Shape	Perimeter	Area
Rectangle	$2 (l + b)$	$l \times b$
Square	$4 s$	$s^{2}$
Triangle	sum of 3 sides	$\frac{1}{2} \times base \times height$

A right-angled triangle's area formula is identical to any triangle's — half of base times height — but its two perpendicular legs conveniently serve as the base and height directly, without needing a separately measured height. A right triangle with legs 9 cm and 12 cm has area $\frac{1}{2} \times 9 \times 12 = 54$ cm², using the legs directly since they are already perpendicular to each other.

6. Circle: circumference and area

A circle's circumference and area both depend only on its radius $r$ , through two fixed formulas that use $π$ (commonly approximated as $\frac{22}{7}$ or 3.14 for calculation):

$Circumference = 2 π r, Area = π r^{2}$

A circle of radius 7 cm (chosen specifically so $\frac{22}{7}$ cancels cleanly) has circumference $2 \times \frac{22}{7} \times 7 = 44$ cm and area $\frac{22}{7} \times 7^{2} = 154$ cm².

7. Volume of a cube and a cylinder

A cube's volume is its side length cubed, since all three dimensions are equal; a cylinder's volume is its circular base area multiplied by its height, combining the circle-area formula from Section 6 with a third dimension.

$Cube: V = s^{3}, Cylinder: V = π r^{2} h$

A cube of side 6 cm has volume $6^{3} = 216$ cm³. A cylinder of radius 7 cm and height 10 cm has volume $\frac{22}{7} \times 7^{2} \times 10 = 1, 540$ cm³.

Worked Examples

Example 1. Solve for $x$ : $7 x - 12 = 3 x + 20$ .

Collecting terms: $7 x - 3 x = 20 + 12$ , so $4 x = 32$ , giving $x = 8$ . Check: $7 (8) - 12 = 44$ and $3 (8) + 20 = 44$ .

Answer: $x = 8$ .

Example 2. If $a = 5$ and $b = 2$ , evaluate $3 a^{2} - 4 b + 7$ .

$3 (25) - 4 (2) + 7 = 75 - 8 + 7 = 74$ .

Answer: 74.

Example 3. Three angles on a straight line are $4 x$ , $2 x$ and $3 x$ . Find the value of $x$ and each angle.

$4 x + 2 x + 3 x = 180 \Rightarrow 9 x = 180 \Rightarrow x = 20$ . The angles are $80°, 40°, 60°$ .

Answer: $x = 20$ ; angles 80°, 40°, 60°.

Example 4. Two angles of a triangle are 72° and 48°. Find the third angle.

$180 - 72 - 48 = 60°$ .

Answer: 60°.

Example 5. A triangle's angles are in the ratio 3:4:5. Find each angle.

Total parts $= 3 + 4 + 5 = 12$ . One part $= 180 \div 12 = 15°$ . The angles are $45°, 60°, 75°$ .

Answer: 45°, 60°, 75°.

Example 6. Find the area and perimeter of a rectangle with length 15 cm and breadth 9 cm.

Area $= 15 \times 9 = 135$ cm². Perimeter $= 2 (15 + 9) = 48$ cm.

Answer: Area = 135 cm², Perimeter = 48 cm.

Example 7. A right-angled triangle has legs of 8 cm and 15 cm. Find its area.

Area $= \frac{1}{2} \times 8 \times 15 = 60$ cm² (the two legs serve directly as base and height).

Answer: 60 cm².

Example 8. Find the circumference and area of a circle with radius 14 cm (use $π = \frac{22}{7}$ ).

Circumference $= 2 \times \frac{22}{7} \times 14 = 88$ cm. Area $= \frac{22}{7} \times 1 4^{2} = \frac{22}{7} \times 196 = 616$ cm².

Answer: Circumference = 88 cm, Area = 616 cm².

Example 9. Find the volume of a cube with side 9 cm.

$9^{3} = 729$ cm³.

Answer: 729 cm³.

Example 10. Find the volume of a cylinder with radius 7 cm and height 15 cm (use $π = \frac{22}{7}$ ).

$V = \frac{22}{7} \times 7^{2} \times 15 = \frac{22}{7} \times 49 \times 15 = 22 \times 7 \times 15 = 2, 310$ cm³.

Answer: 2,310 cm³.

Summary

Linear equations with the variable on both sides collect variable terms to one side and constants to the other, exactly like a one-sided equation applied twice. Evaluating an expression means substituting the given values and then applying BODMAS with no shortcuts.

Angles on a straight line always sum to 180°, angles around a point always sum to 360°, and a triangle's three angles always sum to 180° — all three are fixed facts to apply directly, not values to derive. Angles given as a ratio split the 180° (or 360°) total into parts exactly like ratio-sharing.

Rectangle, square and triangle perimeter/area, and circle circumference/area, all follow fixed formulas built from length, side or radius. Cube volume is side-cubed; cylinder volume is the circular base area multiplied by height — both formulas extend the same area logic into a third dimension.

Key formulas & results

Everything to memorise for the exam hall, in one card. Screenshot this for revision.

Straight line and point angle sums

Angles on a line = 180°, angles around a point = 360°

Fixed geometric facts, not values derived from other information.

Triangle angle sum

∠ A + ∠ B + ∠ C = 180°

True for every triangle regardless of its shape or size.

Rectangle and square

Rectangle: P = 2 (l + b), A = l b Square: P = 4 s, A = s^{2}

l and b are length and breadth; s is the square's side.

Triangle area

A = \frac{1}{2} \times base \times height

For a right triangle, the two perpendicular legs serve directly as base and height.

Circle circumference and area

C = 2 π r, A = π r^{2}

Both depend only on the radius r; use pi = 22/7 or 3.14 as given.

Cube and cylinder volume

Cube: V = s^{3}, Cylinder: V = π r^{2} h

Cylinder volume is the circular base area multiplied by the height.

CUET UG test series

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Algebra, Geometry & Mensuration shows up in every CUET UG paper — sit a timed one and find out where you actually stand.

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Traps CUET UG sets — and how to dodge them

These are the exact option-traps and misreads that cost marks under negative marking.

WATCH OUT

✗ Moving a term across the equals sign without changing its sign

✓ Every term moved from one side of an equation to the other must flip its sign — positive becomes negative and vice versa.

Why it happens: An equation stays balanced only if the same operation is effectively applied to both sides; changing sign on a moved term is what preserves that balance.

WATCH OUT

✗ Using the wrong order of operations when evaluating a substituted expression

✓ After substituting numeric values, apply BODMAS exactly as with any other expression — powers before multiplication, multiplication before addition/subtraction.

Why it happens: Substitution only replaces the variables; it does not change the order in which the resulting arithmetic must be evaluated.

WATCH OUT

✗ Using a measured slanted side as the 'height' in a triangle's area formula

✓ The height must be the PERPENDICULAR distance from the base to the opposite vertex, not any other side.

Why it happens: The area formula (1/2 x base x height) is only valid when height is measured perpendicular to the chosen base.

WATCH OUT

✗ Treating a triangle's ratio-given angles as already the answer, without scaling by the total

✓ Add the ratio's terms to find the total number of parts, divide 180° by that total to find one part's value, then multiply each ratio term by that value.

Why it happens: The ratio terms are only relative proportions, not actual degree values, until scaled against the known 180° total.

WATCH OUT

✗ Confusing a cylinder's volume formula with its surface area formula

✓ Volume is base area times height: pi r^2 h; do not add a separate '2 pi r h' curved-surface term unless the question specifically asks for surface area.

Why it happens: Volume measures the space enclosed (base area x height), a different quantity from surface area, which measures the enclosing surfaces.

Exam-pattern practice

PYQ-style questions with full solutions. Work through them as a readiness check — mark yourself honestly and get your gap report at the end.

Readiness check

Are you exam-ready for Algebra, Geometry & Mensuration?

11 problems from this chapter. Try each one, reveal the worked solution, mark yourself honestly — get your gap report at the end.

11 questions~8 min worth ~250 marks in CUET UG exams

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•A linear equation with the variable on both sides is solved by collecting all variable terms on one side and all constants on the other, flipping sign on every moved term.
•Evaluating a substituted expression still follows BODMAS exactly — powers, then multiplication, then addition/subtraction.
•Angles on a straight line always sum to 180°; angles around a point always sum to 360°; a triangle's three angles always sum to 180°.
•Angles given as a ratio split the known total (180° or 360°) into parts, exactly like ratio-sharing.
•Rectangle: P=2(l+b), A=lb. Square: P=4s, A=s^2. Triangle area = 1/2 x base x height, with a right triangle's legs serving directly as base and height.
•Circle: circumference = 2pir, area = pi*r^2 — both depend only on the radius.
•Cube volume = s^3; cylinder volume = pir^2h (base area x height).

CUET UG question blueprint

How this topic is asked, tier by tier — so you can prep to the pattern.

Typical weightage: Algebra, Geometry & Mensuration contributes an estimated 10-20 of the General Test's 250 marks (about 2-4 of 50 questions)

Question style	Marks each	Typical count	What it tests
Linear equations	5	~1	Solving an equation with the variable on both sides
Evaluating expressions	5	~1	Substituting values and applying BODMAS correctly
Angles on a line or point	5	~1	Applying the 180°/360° angle-sum facts
Triangle angle sum	5	~1	Finding a missing triangle angle from the 180° total
Angles in a ratio	5	~1	Solving for a triangle's angles given as a ratio
Rectangle and square	5	~1	Computing perimeter and area of a rectangle or square
Triangle and right-triangle area	5	~1	Computing a triangle's area, including a right triangle using its legs
Circle	5	~1	Computing a circle's circumference and area
Cube volume	5	~1	Computing a cube's volume
Cylinder volume	5	~1	Computing a cylinder's volume

Prep strategy

Day 1: linear equations with the variable on both sides, and evaluating algebraic expressions.
Day 2: the three fixed angle-sum facts (line, point, triangle) and angles given as a ratio.
Day 3: perimeter/area formulas for rectangle, square, triangle and circle, plus cube and cylinder volume, then mixed timed practice.

Exam-hall strategy

Battle-tested tips from mentors and toppers for this topic under the sectional clock.

Solve equations with the variable on both sides in the same two-step move every time: collect variables, collect constants.
Before computing a triangle's area, confirm which measurement is actually the PERPENDICULAR height, especially when a right triangle's legs are given.
Memorise the circle, cube and cylinder formulas exactly — there is no derivation shortcut to fall back on under time pressure.
For angle-ratio questions, always divide the known total (180° or 360°) by the sum of the ratio terms first, before multiplying out each angle.
Use pi = 22/7 whenever the radius is a multiple of 7, since it cancels to a clean integer answer faster than 3.14 would.

Beyond the exam

Where this skill shows up in the job you're competing for — and in life.

Basic construction and design estimates

Area and volume formulas for rectangles, circles, cubes and cylinders are the everyday calculations behind estimating material, paint, or storage capacity.

Reading structural or technical diagrams

Recognising fixed angle-sum facts is the same reasoning used to read angles in basic technical drawings and floor plans.

Where else this topic is tested

Prepare once, score in every exam that asks it.

SSC CGL Quantitative AptitudeModerate — overlapping mensuration formulas, tested at a somewhat higher difficulty with more shapes

RRB NTPC MathematicsModerate — shares the same basic geometry and mensuration formula set

IBPS PO / SBI PO Quantitative AptitudeLow — mensuration appears only occasionally, algebra even less so

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Questions aspirants ask

Pulled from the Q&A community and mentor sessions.

Roughly 2-4 of the General Test's 50 questions — the smallest of the five Quantitative & Numerical Ability topics, based on pattern analysis of released sample papers.

No trigonometry. Only the fixed angle-sum facts (straight line 180°, point 360°, triangle 180°) and standard perimeter/area/volume formulas for rectangles, squares, triangles, circles, cubes and cylinders are tested.

Add the ratio's terms to get the total number of parts, divide 180° by that total to get one part's value, then multiply each ratio term by that value to get the actual angles.

Use whichever the question specifies — 22/7 is common when the radius is a multiple of 7 (so it cancels cleanly), and 3.14 is common otherwise. Both are acceptable approximations for this level.

The formula is identical (1/2 x base x height), but a right triangle's two perpendicular legs can be used directly as base and height without needing a separately measured perpendicular height.

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Algebra, Geometry & Mensuration

By the end of this chapter you'll be able to…

Before you start — revise these

Algebra, Geometry & Mensuration — CUET UG General Test

1. Linear equations with the variable on both sides

2. Evaluating algebraic expressions

3. Angles on a straight line and at a point

4. Triangle angle sum and angles in a ratio

5. Perimeter and area of standard shapes

6. Circle: circumference and area

7. Volume of a cube and a cylinder

Worked Examples

Summary

Key formulas & results

Traps CUET UG sets — and how to dodge them

Exam-pattern practice

Are you exam-ready for Algebra, Geometry & Mensuration?

5-minute revision

CUET UG question blueprint

Exam-hall strategy

Beyond the exam