Geometry, Measurement & Data

Geometry and the Pythagorean Theorem

Grade 7 · Grade 8 · Grade 9

✓Apply the Pythagorean theorem to solve for unknown side lengths in right-angled triangles.
✓Calculate the surface area and volume of prisms, cylinders, and pyramids.
✓Identify and apply properties of similar figures, including scale factor, to solve problems.
✓Solve real-world problems involving geometry, measurement, and similarity.

Key concepts

Pythagorean Theorem

In any right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (the legs).

a^2 + b^2 = c^2

Right-Angled Triangle

A triangle that has one angle measuring exactly 90 degrees. The sides adjacent to the right angle are called legs, and the side opposite the right angle is called the hypotenuse.

Surface Area

The total area of all the faces (or surfaces) of a three-dimensional object. It is measured in square units.

Varies by shape. For a rectangular prism: SA = 2(lw + lh + wh). For a cylinder: SA = 2πr^2 + 2πrh.

Volume

The amount of space occupied by a three-dimensional object. It is measured in cubic units.

Varies by shape. For a rectangular prism: V = lwh. For a cylinder: V = πr^2h. For a pyramid: V = (1/3)Bh (where B is the area of the base).

Similar Figures

Two figures are similar if they have the same shape but not necessarily the same size. This means their corresponding angles are equal, and the ratio of their corresponding side lengths is constant.

(Side 1 of Figure A) / (Side 1 of Figure B) = (Side 2 of Figure A) / (Side 2 of Figure B) = ... = k

Scale Factor

The ratio of any two corresponding linear dimensions (like side lengths, heights, or radii) of two similar figures. It tells you how many times larger or smaller one figure is compared to the other.

k = (length of a side in new figure) / (length of corresponding side in original figure)

Key facts to remember

1The Pythagorean theorem (a^2 + b^2 = c^2) applies ONLY to right-angled triangles.
2The hypotenuse (c) is always the longest side in a right-angled triangle and is opposite the 90-degree angle.
3Surface area is measured in square units (e.g., cm^2, m^2), while volume is measured in cubic units (e.g., cm^3, m^3).
4Similar figures have identical corresponding angles and proportional corresponding side lengths.
5The scale factor is the ratio of corresponding side lengths in similar figures.
6Formulas for surface area and volume of common 3D shapes (prisms, cylinders, pyramids) are essential.

Worked examples

Example 1

A ladder is 5 metres long and leans against a wall. The base of the ladder is 3 metres away from the wall. How high up the wall does the ladder reach?

IIdentify the right-angled triangle formed by the ladder, the wall, and the ground. The ladder is the hypotenuse (c = 5 m), the distance from the wall is one leg (a = 3 m), and the height up the wall is the other leg (b).

IIWrite the Pythagorean theorem: a^2 + b^2 = c^2.

IIISubstitute the known values: 3^2 + b^2 = 5^2.

IVCalculate the squares: 9 + b^2 = 25.

VIsolate b^2: b^2 = 25 - 9.

VICalculate the difference: b^2 = 16.

VIITake the square root of both sides: b = sqrt(16).

VIIISolve for b: b = 4.

Answer

The ladder reaches 4 metres up the wall.

Always check that the hypotenuse is the longest side.

Example 2

A cylindrical can has a radius of 4 cm and a height of 10 cm. Calculate its surface area and volume. Round your answers to one decimal place.

IIdentify the given values: r = 4 cm, h = 10 cm.

IIRecall the formula for the surface area of a cylinder: SA = 2πr^2 + 2πrh.

IIISubstitute the values: SA = 2π(4)^2 + 2π(4)(10).

IVCalculate the terms: SA = 2π(16) + 2π(40).

VSimplify: SA = 32π + 80π.

VICombine terms: SA = 112π.

VIICalculate the numerical value: SA ≈ 112 * 3.14159 ≈ 351.858.

VIIIRound to one decimal place: SA ≈ 351.9 cm^2.

9Recall the formula for the volume of a cylinder: V = πr^2h.

10Substitute the values: V = π(4)^2(10).

11Calculate: V = π(16)(10).

12Simplify: V = 160π.

13Calculate the numerical value: V ≈ 160 * 3.14159 ≈ 502.654.

14Round to one decimal place: V ≈ 502.7 cm^3.

Answer

The surface area is approximately 351.9 cm^2 and the volume is approximately 502.7 cm^3.

Remember to use the correct units for surface area (cm^2) and volume (cm^3).

Example 3

Triangle ABC is similar to Triangle DEF. If AB = 6 cm, BC = 9 cm, AC = 12 cm, and DE = 4 cm, find the lengths of EF and DF.

IIdentify corresponding sides: AB corresponds to DE, BC corresponds to EF, and AC corresponds to DF.

IICalculate the scale factor (k) using the known corresponding sides: k = DE / AB = 4 cm / 6 cm = 2/3.

IIITo find EF, use the scale factor with BC: EF = k * BC = (2/3) * 9 cm.

IVCalculate EF: EF = 6 cm.

VTo find DF, use the scale factor with AC: DF = k * AC = (2/3) * 12 cm.

VICalculate DF: DF = 8 cm.

Answer

EF = 6 cm and DF = 8 cm.

Ensure you consistently use the same order for the ratio (e.g., small triangle side / large triangle side) when calculating the scale factor and unknown sides.

Common mistakes

✗Applying the Pythagorean theorem to triangles that are not right-angled.
✗Confusing the legs (a, b) with the hypotenuse (c) in the Pythagorean theorem.
✗Mixing up surface area and volume, or using incorrect units for each.
✗Incorrectly identifying corresponding sides or angles when working with similar figures.
✗Forgetting to take the square root at the end of a Pythagorean theorem calculation.

Exam tips

★Always draw a clear diagram for geometry problems, labelling all known and unknown values.
★Show all your steps clearly, especially when using formulas, to earn partial marks even if your final answer is incorrect.
★Double-check your calculations and ensure your final answer includes the correct units.
★Memorize the key formulas for surface area and volume of common 3D shapes, and the Pythagorean theorem.

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