Squares & Square Roots
Match Squares to Values
Draw a line from each square to its value.
Calculate the Square
Circle the correct answer.
4²
6²
8²
9²
12²
Match Square Roots
Draw a line from each square root to its answer.
Perfect Square Sequences
Fill in the missing perfect squares.
Perfect Square or Not?
Sort each number into the correct column.
Estimate Square Roots
Circle the best estimate for each square root.
√20 is between:
√50 is between:
√90 is between:
√130 is between:
Area and Side Length
Use square roots to find the side length of each square.
A square has area 49 cm². What is its side length?
A square has area 144 cm². What is its side length?
A square park has area 625 m². What is its side length?
Square Root — Applied
Match each area to the side length of the square.
Squares and Square Roots — Mixed
Circle the correct answer.
√(4²)
(√25)²
√(49 + 51)
√(36) + √(64)
Square Number Sequences
Fill in the missing terms.
Pythagorean Triples — Right Triangle or Not?
A Pythagorean triple satisfies a² + b² = c². Sort each set of side lengths.
Square Numbers in Context
Use your knowledge of squares and square roots to solve each problem.
A square tile has a side length of 9 cm. What is the area of 12 of these tiles placed side by side in a row?
A square vegetable patch has an area of 196 m². What is the length of its fence?
Is √50 closer to 7 or 8? Explain how you know without a calculator.
Square Root Estimation
Draw a line to match each square root to its correct estimate range.
Between Which Consecutive Integers?
Circle the pair of consecutive integers the square root lies between.
√40
√80
√150
√500
Trial and Improvement for Square Roots
Estimate each square root to 1 decimal place using trial and improvement.
Estimate √5 to 1 decimal place. Show all trials.
Estimate √17 to 1 decimal place. Show all trials.
Pythagoras in Real Life
Use Pythagoras' theorem to solve each problem.
A ladder 10 m long leans against a wall. Its base is 6 m from the wall. How high up the wall does it reach?
A diagonal path cuts across a rectangular park that is 40 m × 30 m. How long is the diagonal path?
Sort by Size of Square Root (Smallest to Largest)
Without calculating, sort these numbers by the size of their square root.
Square Root Rules
Circle the correct simplification.
√(36 × 25) =
√(144/9) =
√(49) × √(4) =
(√8)² =
Explain Square Root Rules
Test and explain each rule.
Is √(a × b) = √a × √b always true? Test with a = 4, b = 9 and a = 4, b = 25.
Is √(a + b) = √a + √b always true? Test with a = 9, b = 16. What do you find?
Order Square Roots — Smallest to Largest
Sort from smallest to largest without using a calculator.
Design a Square Room
Use square roots to solve this design problem.
You want to carpet a square room with an area of 50 m². What is the exact side length? Is this a whole number? How would you express this as a decimal to 1 decimal place?
A square picture frame must have an area of 200 cm². What is the side length to the nearest centimetre?
Simplifying Surds
Simplify each square root by writing it in the form a√b where b has no perfect square factor.
Simplify √12
Simplify √45
Simplify √72
Simplify √98
Surd Comparison
Without a calculator, circle the larger value.
3√2 vs 2√3
√8 vs 3
5 vs 2√6
4√2 vs 3√3
Pythagoras with Surds
Leave your answers as simplified surds where the answer is not a whole number.
A square has area 8 cm². What is the exact side length? Simplify your answer.
A right triangle has legs of 3 cm and 5 cm. Find the exact hypotenuse as a simplified surd.
A right triangle has hypotenuse √50 cm and one leg 5 cm. Find the other leg.
Areas and Perimeters Using Surds
Solve each problem. Leave answers as exact surds or round as directed.
A square has perimeter 20√2 cm. What is its area?
A square has diagonal of length 10 cm. Use d = s√2 to find the side length s and then the area.
Match Surd Equivalence
Which expressions are equal? Sort into equivalent pairs.
Compare Approximation Methods
Use multiple methods to estimate √7.
Method 1 — Trial and Improvement: Find √7 correct to 2 decimal places.
Method 2 — Linear Interpolation: √4 = 2 and √9 = 3. Estimate √7 by proportion: (7−4)/(9−4) = 3/5 of the way from 2 to 3. What estimate does this give?
Which method gave the more accurate estimate? Compare to a calculator value.
Spiral of Theodorus
Build the first six steps of the Spiral of Theodorus.
Draw and label the spiral. Start with a 1×1 right triangle. At each step label the new hypotenuse (√2, √3, √4…). Draw at least 6 triangles.
List the lengths of the hypotenuses after each step: √1, √2, √3, √4, √5, √6.
Investigating Irrational Square Roots
Prove or disprove using evidence.
Use long division or a calculator to find the first 10 decimal places of √2. Does the decimal terminate or repeat?
Without a calculator, explain why √2 cannot equal 1.4142... exactly. (Hint: what is 1.4142² ?)
Hero's Method — Ancient Square Root Algorithm
Hero of Alexandria (c. 50 AD) devised a method to approximate square roots.
Use Hero's method to approximate √10. Start with x₀ = 3. Find x₁, x₂, and x₃. Compare to the calculator value.
Distance Formula — Squares in Coordinate Geometry
The distance between two points (x₁, y₁) and (x₂, y₂) is √((x₂−x₁)² + (y₂−y₁)²).
Find the distance between (0, 0) and (3, 4).
Find the distance between (1, 2) and (4, 6).
A point P = (5, 12). How far is P from the origin?
Create a Pythagorean Triple
Use the formula: if m > n > 0, then (m²−n², 2mn, m²+n²) is a Pythagorean triple.
Use m = 2, n = 1 to generate a Pythagorean triple. Verify that a² + b² = c².
Use m = 3, n = 2 to generate another triple. Verify it.
Use m = 4, n = 1 to generate another triple.
Project: Perfect Squares and Algebra
Investigate algebraic identities involving squares.
Expand (a + b)² = a² + 2ab + b². Verify using a = 3, b = 4: does (3+4)² = 3² + 2(3)(4) + 4²?
Use the identity to calculate 51² without a calculator. Write 51 = 50 + 1 and apply the formula.
Expand (a − b)² = a² − 2ab + b². Verify using a = 7, b = 2: does (7−2)² = 7² − 2(7)(2) + 2²?
Square Numbers in Real Life
Explore square numbers in everyday situations.
- 1Find a square floor tile or wall tile. Measure its side and calculate its area.
- 2Count how many square tiles cover a part of your floor or bathroom wall.
- 3Draw a number line from 0 to 400 and mark all perfect squares up to 400.
- 4Challenge: without a calculator, estimate √150 to one decimal place. Then check with a calculator.
- 5Look up the Spiral of Theodorus and try to draw it on graph paper.