BiologyCBSE Class 11 · Plant Growth and Development

Plant Growth: Phases, Arithmetic and Geometric Growth, Growth Rates

Updated 17 October 2026 · Fact-checked

Plant growth is irreversible and permanent, and is indeterminate because of meristems. It has meristematic, elongation and maturation phases. Arithmetic growth follows Lt = L0 + rt; geometric growth follows W1 = W0 e^(rt), where r is the relative growth rate. Absolute growth rate is the total growth per unit time; relative growth rate is growth per unit initial size per unit time.

In arithmetic growth only one daughter cell keeps dividing, giving a straight line; in geometric growth both divide, so the curve steepens.

Illustration drawn for QuestionBuddy.

Understand Growth Phases and Growth Rates

Growth. A permanent, irreversible increase in size, volume or mass of an organ or the whole organism, usually with metabolic processes (anabolic exceeding catabolic) consuming energy. Plant growth is indeterminate: meristems keep dividing throughout life (open form of growth), unlike animals. Primary growth at apical meristems increases length; secondary growth (lateral meristems) increases girth. Growth is measured by fresh weight, dry weight, length, area, volume and cell number.

Three phases of growth.

Meristematic phase: constantly dividing cells at root and shoot apices; cells are rich in protoplasm, with thin cellulosic walls, large conspicuous nuclei and no vacuoles.
Elongation phase: cells just behind the meristem enlarge, with vacuolation, new cell wall deposition.
Maturation phase: cells attain their maximal size, with wall thickening and protoplasmic modifications.

Growth rate. Increased growth per unit time is the growth rate; it may be arithmetic or geometric.

Arithmetic growth: after mitosis, only one daughter cell continues to divide; the other differentiates and matures. A plot of length against time is a linear curve. Expressed as Lt = L0 + rt, where Lt = length at time t, L0 = length at start, r = growth rate (elongation per unit time). Example: a root elongating at a constant rate.
Geometric growth: both daughter cells retain the ability to divide, and the cell number rises exponentially. Initially slow (lag phase), then rapid (exponential or log phase), then slows when nutrients are limiting (stationary phase); the plot is sigmoid (S-curve). Expressed as W1 = W0 e^(rt), where W1 = final size, W0 = initial size, r = relative growth rate, t = time, e = base of natural logarithms (2.718).

Quantitative comparison.

Absolute growth rate = total growth per unit time.
Relative growth rate = growth in a given time per unit initial parameter; shows the efficiency of growth.

Two leaves can show the same absolute growth, but the smaller leaf has the higher relative growth rate.

Conditions for growth. Water (turgidity for cell enlargement, medium for enzymes), oxygen (for metabolic energy), nutrients (macro and micro), temperature (each organism has an optimum range), and light (for photosynthesis and some processes).

Draw and label: sigmoid growth curve with lag, log and stationary phases.

Growth Phases and Growth Rates: key facts

Arithmetic growth: $L t = L 0 + r t$; Linear
Geometric growth: $W 1 = W 0 e^{(} r t)$; Exponential, r = relative growth rate
Absolute growth rate: Increase in size / time; Per unit time
Relative growth rate: (Increase in size / initial size) / time; Efficiency of growth

How to answer Growth Phases and Growth Rates questions

Questions give sizes over time and ask for length, weight or rates.

1Decide the model: constant increment (arithmetic) or proportional increase (geometric).
2Substitute into the formula and compute step by step.
3For rates, separate absolute (increase) from relative (increase divided by initial).
4Use e = 2.718 and a calculator for exponential values.
5State units clearly.

Growth Phases and Growth Rates in real life

The idea, met in an everyday situation. Answer the question in your own words, then compare with the model answer. The one-line hook is what to remember.

Seedling in a pot

Suppose a bean seedling grows from 5 cm and a stem measuring 0.5 cm per day is recorded for 8 days, while a bacterial culture doubles every hour in nutrient broth.

Which follows arithmetic and which geometric growth, and what is the seedling length?

Try it first, then show the solutionSolution

The stem with a constant increment is arithmetic.
Length = 5 + 0.5 x 8 = 9 cm.
The culture, whose cells all divide, is geometric (exponential at first).

Answer: Stem: arithmetic, 9 cm; culture: geometric.

Memory hook: Constant steps versus compounding.

Tricks to remember Growth Phases and Growth Rates

Short memory aids. Say each one aloud once, then use it on a question the same day.

LRT
Length at time = starting length + rate x time.
Use it when: Arithmetic
W-W-e
W1 = W0 e^(rt) compounds.
Use it when: Geometric
M-E-M
Meristematic, Elongation, Maturation.
Use it when: Phases

Growth Phases and Growth Rates: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Arithmetic growth · 2 marks · Very short answerBoard level

A shoot is 12 cm long and elongates at a steady 1.5 cm per day. Find its length after 6 days using the formula.

Try it first, then show the solutionSolution

Lt = L0 + rt (1 mark).
Lt = 12 + 1.5 x 6 = 12 + 9 = 21 cm (1 mark).

Answer: 21 cm.

Example 2: Phases · 3 marks · Short answerBoard level

Describe the three phases of growth with cell features in each.

Try it first, then show the solutionSolution

Meristematic: dividing cells with thin walls, dense protoplasm, large nuclei (1 mark).
Elongation: vacuolation, cell enlargement and new wall deposition (1 mark).
Maturation: wall thickening and protoplasmic modification (1 mark).

Answer: Divide, enlarge, mature.

Example 3: Relative rates · 4 marks · Short answerHarder board level

Leaf A grows from 4 cm2 to 8 cm2 and leaf B from 40 cm2 to 48 cm2 in the same time. Find the absolute growth and the relative growth (per cent) of each, and say which is growing more efficiently.

Try it first, then show the solutionSolution

Absolute: A = 8 - 4 = 4 cm2; B = 48 - 40 = 8 cm2 (1 mark).
Relative: A = 4/4 x 100 = 100 per cent (1 mark).
Relative: B = 8/40 x 100 = 20 per cent (1 mark).
A has the lower absolute but higher relative growth, so it grows more efficiently (1 mark).

Answer: A: 4 cm2, 100%; B: 8 cm2, 20%; A is more efficient.

Example 4: Geometric growth · 3 marks · Short answerHarder board level

A culture has initial mass 2 g and relative growth rate 0.1 per day. Find its mass after 5 days using W1 = W0 e^(rt) (take e^0.5 = 1.6487).

Try it first, then show the solutionSolution

rt = 0.1 x 5 = 0.5 (1 mark).
W1 = 2 x e^0.5 = 2 x 1.6487 (1 mark).
W1 = 3.2974, about 3.30 g (1 mark).

Answer: About 3.30 g.

Example 5: Comparing curves · 5 marks · Long answerHOTS / competency

Why does the S-curve of a plant cell population differ from the linear curve of a root elongating at constant rate? Link to cell behaviour after mitosis and name the three phases of the S-curve.

Try it first, then show the solutionSolution

In arithmetic growth only one daughter cell divides again; the other matures, so increments stay constant (1 mark).
In geometric growth both daughters divide, so numbers compound (1 mark).
Resources limit growth later, causing the plateau (1 mark).
Phases: lag, exponential (log) and stationary (1 mark).
Linear growth plot: straight line; geometric: sigmoid (1 mark).

Answer: Both daughters dividing gives exponential then stationary growth.

Where students lose marks in Growth Phases and Growth Rates

The slips examiners see most often, and what to write instead.

Mistake: Relative equals absolute
Terms mixed.
Wrong working
Relative growth is the increase in size.
Right working
Relative growth is increase divided by initial size.
Do this: Divide by initial.
Usually costs: 1 mark.
Mistake: e = 10
Constant mistake.
Wrong working
Use e = 10.
Right working
e = 2.718.
Do this: Natural base.
Usually costs: 1 mark.
Mistake: Arithmetic growth S-curve
Curve mixed.
Wrong working
Arithmetic growth is S-shaped.
Right working
Arithmetic is linear; geometric is S-shaped.
Do this: Linear.
Usually costs: 1 mark.

Exam tips

Show formula, substitution and units.
Convert relative growth to per cent only if asked.
Sketch the S-curve with three phases.
List the five conditions for growth.

Practice questions on Growth Phases and Growth Rates

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
The formula Lt = L0 + rt describes
(a) geometric growth (b) arithmetic growth (c) relative growth (d) differentiation
Hint
Constant increment.
Answer
(b).
Q2 · 1 mark · Assertion-reasonHarder board level
Assertion (A): Relative growth rate indicates the efficiency of growth.
Reason (R): It expresses growth per unit of initial size.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Hint
Per unit initial.
Answer
(a).
Q3 · 2 marks · Very short answerHarder board level
A seedling grows from 12 cm to 18 cm in 3 days. Find its absolute growth rate per day and the relative growth rate per day as a percentage of initial length.
Hint
Compute both.
Answer
Absolute = (18 - 12)/3 = 2 cm per day. Relative = 2/12 x 100 = 16.7 per cent per day.
Q4 · 4 marks · Case-basedHarder board level
Read and answer.
Suppose a graph shows a slow rise for 2 days, a steep rise for 6 days, then a plateau.
(a) Name the type of growth. (1)
(b) Name the three phases. (1)
(c) Give a reason for the plateau. (2)
Hint
S-curve.
Answer
(a) Geometric (sigmoid). (b) Lag, exponential and stationary. (c) Nutrients and space become limiting.
Q5 · 3 marks · Short answerHOTS / competency
Suppose a plant is deprived of oxygen but given water and light. Predict growth in a root tip and justify with the requirements for growth.
Hint
Energy for growth.
Answer
Root tip growth would fall: oxygen is needed for aerobic respiration to supply energy for cell division and enlargement, so growth would stop or slow despite water and light.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.What are the phases of growth?Show answer
Meristematic, elongation, maturation.
2.What is r in W0 e^(rt)?Show answer
Relative growth rate.
3.Growth form of plants?Show answer
Open (indeterminate).
4.What is e?Show answer
2.718.
5.What limits the end of the S-curve?Show answer
Nutrient supply.
6.Which phase has vacuolation?Show answer
Elongation.

Growth Phases and Growth Rates: frequently asked questions

What is growth?

Permanent increase in size.

What is arithmetic growth?

Constant increase per unit time.

What is geometric growth?

Growth proportional to current size.

What is open growth?

Continued growth by meristems.

How is growth measured?

Length, weight, area, volume or cell number.