SSC CGL Maths · Mensuration 3D

Prisms and Pyramids

A prism holds its base area times its height, and its side faces together are the base perimeter times the height. A pyramid on the same base and height holds one third as much.

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Why this matters

Twenty-three questions in the bank, sixteen EASY and seven MODERATE, and every one of them comes from the 2025 shifts; the 2023 and 2024 shifts in the bank have none. Most rows are one line once you know two facts, base area times height for a prism and one third of that for a pyramid. The MODERATE ones stack layers whose heights form an AP, which is one addition more.

Concept 1 of 4: Volume of a prism

A prism is one shape stacked straight up: every slice parallel to the base is a copy of the base. So the volume is the base area carried through the height, exactly as for a cuboid, which is the prism with a rectangle for its base.

Definition

Volume of a right prism $=$ base area $\times$ height.
Base areas you will need: an equilateral triangle of side $a$ has area $\frac{\sqrt{3}}{4}a^2$ ; a right triangle has $\frac{1}{2} \times$ the two shorter sides; a trapezium has $\frac{1}{2} \times$ (sum of the parallel sides) $\times$ the distance between them; a regular hexagon of side $a$ is six equilateral triangles, $\frac{3\sqrt{3}}{2}a^2$ .
With the base unchanged, the volume changes in step with the height: $1.5$ times the height gives $1.5$ times the volume.
Two prisms of equal height compare as their base areas.
A composite tank (one prism on another): find each volume in one unit and add.

Prism and two base areas

V = \text{base area} \times h \qquad A_{\triangle} = \frac{\sqrt{3}}{4}a^2 \qquad A_{\text{hexagon}} = \frac{3\sqrt{3}}{2}a^2

Worked example

A right prism has an equilateral triangle of side 8 cm as its base and is 5 cm tall. Find its volume.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 13 Sep 2025, Shift 3 · Q72Easy

Example 1 · Mensuration 3D · Prisms and Pyramids

If the base of a prism is a regular hexagon of side

6

cm and the height is

10

cm, what is its volume?

A hexagon is six triangles

A regular hexagon of side

a

splits into six equilateral triangles of side

a

, so its area is

6 \times \frac{\sqrt{3}}{4}a^2 = \frac{3\sqrt{3}}{2}a^2

. Writing

\frac{\sqrt{3}}{4}a^2

alone gives one triangle.

Square centimetres times metres

A base of

60\ \text{cm}^2

and a height of 1.2 m give

60 \times 120 = 7200\ \text{cm}^3

, not

72

. Change the height to centimetres first.

Drill 9 more on volume of a prism

Concept 2 of 4: Lateral surface area of a prism

The side faces of a prism are rectangles, each as tall as the prism. Laid side by side they form one long rectangle whose length is the distance round the base.

Definition

Lateral (side) surface $=$ perimeter of the base $\times$ height.
Total surface $=$ lateral surface $+ 2 \times$ base area.
The lateral surface needs only the perimeter. A question that also gives the base area may be testing whether you know that.
Base sides given as a ratio with a known perimeter: the lateral surface still needs only the perimeter.

Lateral and total surface of a prism

\text{LSA} = \text{perimeter of base} \times h \qquad \text{TSA} = \text{LSA} + 2 \times \text{base area}

Worked example

A right prism has a triangular base with sides 8 cm, 15 cm and 17 cm, and its height is 10 cm. Find its lateral surface area and its total surface area.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 26 Sep 2025, Shift 1 · Q63Easy

Example 2 · Mensuration 3D · Prisms and Pyramids

A triangular prism has base side lengths 5 cm, 6 cm, and 7 cm and height 10 cm. What is its lateral surface area?

Base area in place of perimeter

The lateral surface multiplies the perimeter of the base by the height. Base area times height is the volume, a different quantity.

Drill 3 more on lateral surface area of a prism

Concept 3 of 4: A prism built in layers whose heights form an AP

Every layer sits on the same base, so the whole stack is one prism whose height is all the layer heights added together. The only real work is adding an AP.

Definition

Same base in every layer, so volume $=$ base area $\times$ (sum of the layer heights).
For $n$ layers running from height $a$ to height $l$ : sum $= \frac{n}{2}(a + l)$ .
If you know the first height $a$ and the step $d$ , the last height is $a + (n - 1)d$ .
Rising or falling does not matter: only the total height counts.
The same rule works for a stack of cylinders of one radius.

Sum of the layer heights

H = \frac{n}{2}(a + l) \qquad l = a + (n - 1)d \qquad V = \text{base area} \times H

Worked example

A prism with a square base of side 3 cm is built in 6 layers. The bottom layer is 2 cm tall and each layer is 3 cm taller than the one below it. Find the volume.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 20 Sep 2025, Shift 1 · Q65Moderate

Example 3 · Mensuration 3D · Prisms and Pyramids

A prism with a square base of side length 5 cm is constructed in layers. Its height follows an arithmetic progression, increasing from 4 cm in the first layer to 16 cm in the fourth layer. What is the total volume of the prism?

Using one layer's height

The prism is as tall as all its layers together. Base area times the tallest layer, or times the average layer without multiplying by the number of layers, gives only part of the volume.

Drill 3 more on a prism built in layers whose heights form an ap

Concept 4 of 4: Volume and lateral surface of a pyramid

A pyramid on the same base and with the same height as a prism holds exactly one third of it. Its side faces are triangles, each as tall as the slant height.

Definition

Volume $= \frac{1}{3} \times$ base area $\times$ height. The height is vertical: from the apex straight down to the base.
Lateral surface of a regular pyramid $= \frac{1}{2} \times$ perimeter of the base $\times$ slant height (the height of each triangular face).
For a square base of side $a$ this is $2al$ .
A pyramid set on a cube or a prism: add the two volumes.

Pyramid

V = \frac{1}{3} \times \text{base area} \times h \qquad \text{LSA} = \frac{1}{2} \times \text{perimeter} \times l

$h$ vertical height, apex to base
$l$ slant height of a face

Worked example

A pyramid has a square base of side 6 cm and a vertical height of 4 cm. The slant height of each face is 5 cm. Find its volume and its lateral surface area.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 21 Sep 2025, Shift 1 · Q58Easy

Example 4 · Mensuration 3D · Prisms and Pyramids

A company designs a new chocolate box in the shape of a regular right pyramid with a square base. The base side of the box is 10 cm and its height is 12 cm. Due to packaging constraints, the box can only be filled up to

90\%

of its total volume. If the company wants to estimate the total cost of chocolate, knowing that 1

\text{cm}^3

of chocolate costs ₹0.50, what is the cost of chocolate to fill one such box?

Forgetting the one third

A pyramid is one third of the prism on the same base and height. Base area times height alone gives three times the answer.

Slant height in the volume

A square pyramid of base side 6 cm can have vertical height 4 cm and slant height 5 cm. Use 4 for the volume and 5 for the faces.

Drill 4 more on volume and lateral surface of a pyramid

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (4)

Volume of a prism
Prism and two base areas
$V = \text{base area} \times h \qquad A_{\triangle} = \frac{\sqrt{3}}{4}a^2 \qquad A_{\text{hexagon}} = \frac{3\sqrt{3}}{2}a^2$
Lateral surface area of a prism
Lateral and total surface of a prism
$\text{LSA} = \text{perimeter of base} \times h \qquad \text{TSA} = \text{LSA} + 2 \times \text{base area}$
A prism built in layers whose heights form an AP
Sum of the layer heights
$H = \frac{n}{2}(a + l) \qquad l = a + (n - 1)d \qquad V = \text{base area} \times H$
Volume and lateral surface of a pyramid
Pyramid
$V = \frac{1}{3} \times \text{base area} \times h \qquad \text{LSA} = \frac{1}{2} \times \text{perimeter} \times l$

Watch out for (6)

A hexagon is six triangles
→ Volume of a prism
Square centimetres times metres
→ Volume of a prism
Base area in place of perimeter
→ Lateral surface area of a prism
Using one layer's height
→ A prism built in layers whose heights form an AP
Forgetting the one third
→ Volume and lateral surface of a pyramid
Slant height in the volume
→ Volume and lateral surface of a pyramid

Test yourself on Mensuration 3D

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