4.3 Geometry & Measurement

Perimeter, Area & Circles

Perimeter is the distance around a flat shape (add the sides); area is the space inside it (square units). Memorize this core table — it appears constantly on placement tests.

For circles, r is the radius (center to edge) and the diameter d = 2r. Use π ≈ 3.14 unless told to leave the answer "in terms of π."

Worked example (rectangle). A garden is 12 m by 5 m. Area = 12 × 5 = 60 m²; perimeter = 2(12) + 2(5) = 24 + 10 = 34 m.

Worked example (circle). A circle has radius 6 cm. Area = π(6)² = 36π ≈ 113.1 cm²; circumference = 2π(6) = 12π ≈ 37.7 cm.

Worked example (triangle). A triangle has base 10 in and height 7 in. Area = ½ × 10 × 7 = 35 in². The height must be perpendicular to the chosen base.

The Pythagorean Theorem & Solids

In a right triangle, the Pythagorean theorem relates the two legs a and b to the hypotenuse c (the side opposite the right angle):

a² + b² = c²

The hypotenuse is always the longest side and always sits alone on the c side of the equation.

Worked example. Legs 6 and 8; find the hypotenuse.

• c² = 6² + 8² = 36 + 64 = 100 → c = √100 = 10.

Worked example (solving for a leg). Hypotenuse 13, one leg 5; find the other leg.

• 5² + b² = 13² → 25 + b² = 169 → b² = 144 → b = 12.

For three dimensions, volume measures the space inside a solid (cubic units) and surface area the total outside skin.

Worked example (cylinder). A can has radius 3 cm and height 10 cm. Volume = π(3)²(10) = 90π ≈ 282.7 cm³.

Angles & Similar Triangles

Two angle relationships show up repeatedly:

• Complementary angles add to 90°. If one is 35°, its complement is 55°.
• Supplementary angles add to 180°. If one is 110°, its supplement is 70°.

Also useful: the three interior angles of any triangle sum to 180°, and angles in a straight line sum to 180°.

Worked example. Two complementary angles are x and 2x. Then x + 2x = 90 → 3x = 90 → x = 30°, so the angles are 30° and 60°.

Similar triangles have the same shape but different size: corresponding angles are equal and corresponding sides are proportional. To find a missing side, set up a ratio of corresponding sides and cross-multiply.

Worked example. Triangle ABC is similar to triangle DEF. Side AB = 4 corresponds to DE = 6, and BC = 5 corresponds to EF = x. Set up the proportion:

• 4/6 = 5/x → 4x = 30 → x = 7.5.

This is the basis of indirect measurement — e.g., using a shadow and a known height to find a flagpole's height.

Common mistake: Using diameter instead of radius in πr² — if a circle's diameter is 10, the radius is 5, so area = π(5)² = 25π, not π(10)². Another trap: in the Pythagorean theorem, plugging the hypotenuse in for a leg. The longest side is always c.

Unit Conversion & Composite Figures

Placement tests mix in measurement conversions. Keep these handy: 1 ft = 12 in, 1 yd = 3 ft, 1 m = 100 cm, 1 km = 1000 m. When converting areas, the factor is squared, and for volumes it is cubed — a 1 ft² tile is 12 × 12 = 144 in², not 12 in².

Worked example. A floor is 4 ft by 3 ft. Its area is 12 ft². In square inches that is 12 × 144 = 1,728 in².

Composite figures are built from simpler shapes; find the area by adding or subtracting pieces.

Worked example. A 10 m by 6 m rectangle has a semicircle of diameter 6 m attached to one short end. Rectangle area = 60 m². The semicircle has radius 3, so its area = ½ · π(3)² = 4.5π ≈ 14.1 m². Total ≈ 74.1 m².

Worked example (shaded region). A circle of radius 4 is inscribed in a square of side 8. The leftover (corner) area is the square minus the circle: 8² − π(4)² = 64 − 16π ≈ 64 − 50.3 = 13.7 square units.

When a problem asks for a shaded portion, identify whether you are adding shapes or subtracting a hole, and keep units consistent throughout. Round only at the final step to avoid accumulating error, and label your answer with the correct square or cubic units.