Build your problem-solving skills with this free PMMA geometry reviewer and practice test. Geometry questions ask you to work with shapes, angles, lengths, areas, and volumes using reliable formulas and careful reasoning.
This page reviews the core geometry topics, lists the formulas you need most, and walks through worked examples for each. These are independently prepared learning materials, not official Philippine Merchant Marine Academy examination questions. Use these exercises as supplementary Mathematics practice, rather than as a guaranteed preview of the actual examination.
PMMA Geometry Practice Test
Sketch and label a figure for each problem before choosing an answer. A quick drawing often reveals the relationship you need.
Note the units in each problem and make sure your answer uses the same ones.
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Geometry Basics
Geometry studies the size, shape, and position of figures. A few basic terms appear in almost every problem:
- A point marks an exact location and has no size.
- A line extends without end in both directions, a ray extends without end in one direction, and a line segment has two endpoints.
- An angle is formed by two rays that share an endpoint, called the vertex.
Angles are classified by their measure:
| Angle type | Measure |
|---|---|
| Acute | less than 90° |
| Right | exactly 90° |
| Obtuse | more than 90° but less than 180° |
| Straight | exactly 180° |
| Reflex | more than 180° but less than 360° |
Geometry Topics to Practice
1. Angle Relationships
Some pairs of angles have fixed relationships:
- Complementary angles add up to 90°.
- Supplementary angles add up to 180°.
- Vertical angles, formed by two intersecting lines, are equal.
When two parallel lines are cut by a transversal, corresponding angles are equal, alternate interior angles are equal, and same-side interior angles are supplementary.
Example: Two angles are supplementary, and one measures 40° more than the other. Find both angles.
- Let the smaller angle be x: x + (x + 40) = 180
- Solve: 2x = 140, so x = 70
- The angles are 70° and 110°
2. Triangles
The three interior angles of any triangle add up to 180°. An exterior angle equals the sum of the two interior angles that are not next to it.
Triangles are classified by sides as equilateral (three equal sides), isosceles (at least two equal sides), or scalene (no equal sides). They are classified by angles as acute, right, or obtuse.
Example: The angles of a triangle are in the ratio 1 : 2 : 3. Find each angle.
- Write the angles as x, 2x, and 3x: x + 2x + 3x = 180
- Solve: 6x = 180, so x = 30
- The angles are 30°, 60°, and 90°
3. The Pythagorean Theorem
In a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: a² + b² = c². The hypotenuse, c, is always the side opposite the right angle.
Example: A ship sails 6 km north and then 8 km east. How far is it from its starting point?
- The two paths form the legs of a right triangle: 6² + 8² = c²
- Compute: 36 + 64 = 100, so c = √100
- The ship is 10 km from its starting point
Memorizing common Pythagorean triples saves time: 3–4–5, 5–12–13, 8–15–17, and 7–24–25. Multiples such as 6–8–10 also work.
Two special right triangles appear often. In a 45°–45°–90° triangle, the sides are in the ratio 1 : 1 : √2. In a 30°–60°–90° triangle, the sides are in the ratio 1 : √3 : 2.
4. Similar Triangles
Similar triangles have the same shape but not necessarily the same size. Their corresponding angles are equal, and their corresponding sides are proportional.
Example: A 2-meter pole casts a 3-meter shadow. At the same time, a ship’s mast casts a 15-meter shadow. How tall is the mast?
- Set up a proportion of height to shadow: h ÷ 15 = 2 ÷ 3
- Solve: h = 15 × 2 ÷ 3
- The mast is 10 meters tall
5. Polygons
The interior angles of a polygon with n sides add up to (n − 2) × 180°. In a regular polygon, all sides and angles are equal, so each interior angle is that sum divided by n. The exterior angles of any convex polygon add up to 360°.
Example: Find each interior angle of a regular hexagon.
- Sum of interior angles: (6 − 2) × 180° = 720°
- Each angle: 720° ÷ 6 = 120°
Example: Each exterior angle of a regular polygon measures 30°. How many sides does it have?
- Divide the total of the exterior angles by one angle: 360° ÷ 30° = 12 sides
6. Circles
The radius, r, runs from the center to the circle. The diameter, d, passes through the center and equals 2r.
- Circumference: C = 2πr = πd
- Area: A = πr²
- Arc length: (central angle ÷ 360°) × 2πr
- Sector area: (central angle ÷ 360°) × πr²
An inscribed angle measures half of the central angle that intercepts the same arc.
Example: Find the circumference and area of a circle with a radius of 7 cm. Use π ≈ 22/7.
- Circumference: 2 × 22/7 × 7 = 44 cm
- Area: 22/7 × 7² = 22/7 × 49 = 154 cm²
7. Perimeter and Area
Perimeter is the distance around a figure, measured in units such as cm. Area is the space inside it, measured in square units such as cm².
| Figure | Perimeter | Area |
|---|---|---|
| Square (side s) | 4s | s² |
| Rectangle (length l, width w) | 2l + 2w | lw |
| Triangle (base b, height h) | sum of the three sides | ½bh |
| Parallelogram (base b, height h) | sum of the four sides | bh |
| Trapezoid (bases b₁ and b₂, height h) | sum of the four sides | ½(b₁ + b₂)h |
| Circle (radius r) | 2πr | πr² |
Example: A trapezoid has bases of 8 cm and 12 cm and a height of 5 cm. Find its area.
- Apply the formula: ½ × (8 + 12) × 5
- Area: ½ × 20 × 5 = 50 cm²
8. Volume and Surface Area
Volume measures the space inside a solid, in cubic units such as m³. Surface area is the total area of its outer faces, in square units.
| Solid | Volume | Surface area |
|---|---|---|
| Cube (edge s) | s³ | 6s² |
| Rectangular prism (l, w, h) | lwh | 2(lw + lh + wh) |
| Cylinder (radius r, height h) | πr²h | 2πr² + 2πrh |
| Cone (radius r, height h, slant height l) | ⅓πr²h | πr² + πrl |
| Sphere (radius r) | ⁴⁄₃πr³ | 4πr² |
| Pyramid (base area B, height h) | ⅓Bh | B + area of the triangular faces |
Example: A cylindrical fuel tank has a radius of 2 m and a height of 5 m. Find its volume.
- Apply the formula: π × 2² × 5 = 20π
- Volume: 20π m³, or about 62.8 m³ using π ≈ 3.14
Since 1 m³ equals 1,000 liters, the tank holds about 62,800 liters.
9. Coordinate Geometry
Coordinate geometry places figures on the xy-plane. For two points (x₁, y₁) and (x₂, y₂):
- Distance: d = √[(x₂ − x₁)² + (y₂ − y₁)²]
- Midpoint: [(x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2]
- Slope: m = (y₂ − y₁) ÷ (x₂ − x₁)
Example: Find the distance, midpoint, and slope for the points (1, 2) and (7, 10).
- Distance: √[(7 − 1)² + (10 − 2)²] = √(36 + 64) = 10
- Midpoint: [(1 + 7) ÷ 2, (2 + 10) ÷ 2] = (4, 6)
- Slope: (10 − 2) ÷ (7 − 1) = 8 ÷ 6 = 4/3
How to Solve Geometry Questions
Draw and label a figure. Even a rough sketch helps you see which sides, angles, and relationships are involved. Mark every given measurement on it.
Write the formula before substituting. Writing it out first reduces errors, especially with formulas that include ½, ⅓, or π.
Convert units first. If one length is in meters and another in centimeters, convert them to the same unit before calculating.
Look for hidden shapes. Many problems contain a right triangle, a pair of parallel lines, or similar triangles that are not pointed out directly.
Check how the choices are written. Some answers use π ≈ 3.14, others use 22/7, and others leave π in the answer. Match your work to the form of the choices.
Common Mistakes to Avoid
One frequent mistake is using the diameter in place of the radius. If a problem gives a diameter of 10 cm, the radius is 5 cm, and the area is 25π cm², not 100π cm².
Another mistake is applying the Pythagorean theorem to triangles that are not right triangles, or treating a leg as the hypotenuse. The hypotenuse is always the longest side.
Learners also drop the fractions in formulas. A triangle’s area needs ½, and the volume of a cone or pyramid needs ⅓.
Finally, keep perimeter, area, and volume separate. Perimeter uses plain units, area uses square units, and volume uses cubic units. An answer with the wrong unit is usually the wrong measurement.
Make Geometry Practice Part of Your Review
Copy the formulas from this reviewer onto a single sheet and review them until you can recall each one without looking. Then solve a few problems by hand every day, drawing a figure for each.
Record the types of problems you miss most often, such as circles or similar triangles, and practice those until the method feels natural.
For additional English, Mathematics, Science, and reasoning exercises, visit our PMMA entrance exam reviewer and coverage guide.