3D Figures, Heights and Distances Notes for XAT: Concepts, Formulas, Worked Examples & Practice

3D Figures, Heights and Distances notes for XAT: 23 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Chapter Roadmap: 3D Figures, Heights and Distances

Chapter Journey

1. 3D Solids & Surface Area

Mastering volumes, surface areas, frustums, combinations of solids, and shortest paths on 3D surfaces.

2. Heights, Distances & Angles of Elevation

Applying trigonometric ratios to solve for unknown heights and horizontal distances.

The Shift to Three Dimensions

The Core Distinction

When moving from 2D to 3D geometry, we shift our focus from flat boundaries to solid spaces.

Volume

The 3D space enclosed. Measured in cubic units ( $c m^{3}$ , $m^{3}$ ).

Surface Area

The 2D area covering the exterior. Measured in square units ( $c m^{2}$ , $m^{2}$ ).

Two Types of Surface Area

Curved / Lateral (CSA / LSA): Only the side walls.
Total (TSA): Side walls PLUS all top and bottom bases.

Core 3D Solids and Their Measures

Solid	Volume	CSA / LSA	TSA
Cube ( $a$ )	$a^{3}$	$4 a^{2}$	$6 a^{2}$
Cuboid ( $l, b, h$ )	$l bh$	$2 h (l + b)$	$2 (l b + bh + h l)$
Cylinder ( $r, h$ )	$π r^{2} h$	$2 π r h$	$2 π r (r + h)$
Cone ( $r, h, l$ )	$\frac{1}{3} π r^{2} h$	$π r l$	$π r (r + l)$
Sphere ( $r$ )	$\frac{4}{3} π r^{3}$	N/A	$4 π r^{2}$
Hemisphere ( $r$ )	$\frac{2}{3} π r^{3}$	$2 π r^{2}$	$3 π r^{2}$

Note for Cone: Slant height $l = r^{2} + h^{2}$ .

The Frustum of a Cone

A frustum is formed by cutting the top of a cone with a plane parallel to its base. Let $R$ = larger base radius, $r$ = smaller top radius, $h$ = vertical height.

Slant Height ( $L$ )

$L = h^{2} + (R - r)^{2}$

Volume

$V = \frac{1}{3} π h (R^{2} + r^{2} + R r)$

CSA

$C S A = π L (R + r)$

TSA

$T S A = π L (R + r) + π R^{2} + π r^{2}$

3D Figures, Heights and Distances: Solved Questions with Step-by-Step Explanations (2 Problems)

Question 1 · Quantitative Aptitude and Data Interpretation (QA & DI) MCQ

According to the final exam checklist for heights and distances, which trigonometric ratio is highlighted as the "primary tool" because it directly links height and horizontal distance for most problems?

A.
Sine
B.
Cosine
C.
Secant
D.
Tangent

Correct Answer:

Step-by-Step Solution

Key idea: This is a direct recall question from the final exam checklist.

Step 1: Review the final exam checklist for Heights and Distances.

Step 2: The checklist explicitly states: "Tangent is King: Use $tan θ = h / d$ for 90% of problems."

Step 3: This is because Tangent directly relates the opposite side (height) to the adjacent side (horizontal distance), which are the two most commonly given or requested variables.

Answer: Tangent.

Question 2 · Quantitative Aptitude and Data Interpretation (QA & DI) MCQ

According to the final exam checklist for heights and distances, what is the recommended very first step when approaching any problem in this topic?

A.
Calculate the tangent of the given angle
B.
Draw a horizontal line from the observer's eye
C.
Add the observer's height to the total
D.
Assume the angles are complementary

Correct Answer:

Step-by-Step Solution

Key idea: This is a direct recall question from the final exam checklist.

Step 1: Review the final exam checklist for Heights and Distances.

Step 2: The checklist explicitly states as step 1: "Draw the Horizontal: Always start by drawing a horizontal line from the observer's eye."

Step 3: This is crucial because it establishes the right angle and the reference line for the angle of elevation or depression.

Answer: Draw a horizontal line from the observer's eye.

More notes in this unit

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3D Figures, Heights and Distances Notes for XAT: Concepts, Formulas, Worked Examples & Practice

3D Figures, Heights and Distances notes for XAT: 23 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

A question from this chapter

Question 1

According to the final exam checklist for heights and distances, which trigonometric ratio is highlighted as the "primary tool" because it directly links height and horizontal distance for most problems?

Question 2

According to the final exam checklist for heights and distances, what is the recommended very first step when approaching any problem in this topic?

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3D Figures, Heights and Distances Notes for XAT: Concepts, Formulas, Worked Examples & Practice

Chapter Roadmap: 3D Figures, Heights and Distances

Chapter Journey

1. 3D Solids & Surface Area

2. Heights, Distances & Angles of Elevation

The Shift to Three Dimensions

The Core Distinction

Volume

Surface Area

Two Types of Surface Area

Core 3D Solids and Their Measures

Core 3D Solids and Their Measures

The Frustum of a Cone

The Frustum of a Cone

Slant Height (L)

Volume

CSA

TSA

3D Figures, Heights and Distances: Solved Questions with Step-by-Step Explanations (2 Problems)

Step-by-Step Solution

Step-by-Step Solution

More notes in this unit

A question from this chapter

Login to view the complete notes

Personalised first. High quality throughout.

Built around you, not around a syllabus PDF

Revision that hits your weak spots

Questions calibrated to the real exam

Notes written for recall, not for volume

One place for everything

Honest progress

Unlock the whole course

Slant Height ( $L$ )