Surface Area and Volume UPCAT Reviewer: Prisms, Cylinders, Cones, and Spheres

TEACHER ABI UPCAT MATHEMATICS

Surface Area and Volume

Choose between capacity and material coverage, include only the surfaces actually present, and solve multi-step solid-geometry applications.

5-10 minute lesson27 original questionsAdaptive practiceSaves progress

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Surface Area and Volume

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Volume fills; surface area covers

Volume measures the space inside a solid and uses cubic units. Surface area measures the total exposed covering and uses square units.

Before using a formula, decide which faces or bases actually exist. An open container has the same volume formula but less surface area than a closed one.

Prism

V = Bh; rectangular prism V = lwh

Cylinder

V = πr²h; closed SA = 2πr² + 2πrh

Pyramid or cone

V = (13)Bh

Sphere

SA = 4πr²; V = (43)πr³

Scaling

Length ×k, surface area ×k², volume ×k³.

DO IT FAST

List the surfaces before adding their areas

1. Ask what is being measured. Filling and capacity mean volume; wrapping, painting, and material mean surface area.

2. Sketch or list every exposed face. Cross out any missing top or excluded base.

3. Keep units consistent. Area uses squared units; volume uses cubed units.

4. Check radius versus diameter.

Why it works

This prevents the most common solid-geometry error: applying the full closed-surface formula to a container that is open or only partly covered.

WORKED EXAMPLES

Five forms you should recognize

1. Rectangular storage box

V = 10(6)(4) = 240 cm³SA = 2[10(6) + 10(4) + 6(4)] = 248 cm²

2. Closed cylinder

SA = 2πr² + 2πrh = 66π cm²

3. Open-top box

Material = bottom + four walls. Do not add a 12×5 top.

4. Sphere

V = (43)π(6³) = 288π m³

COMMON TRAPS

Check before you commit

• Confusing square and cubic units
• Using diameter in place of radius
• Including a missing top in surface area
• Using total surface area when only the curved side is covered
• Forgetting the one-third factor for cones and pyramids
• Scaling volume by k² instead of k³

FIVE-FORM SKILL CHECK

Do you need the lesson-or just practice?

One original question in each form recommends your next step. It does not yet verify mastery.

CHOOSE YOUR PRACTICE

Work at the level you need.

Foundations

Build the core procedure with immediate explanations.

Core Practice

Use mixed forms with less scaffolding.

UPCAT-Style Transfer

Apply the competency in unfamiliar representations.

FRESH MASTERY CHECK

Ready to verify this competency?

A score of 5/5 verifies mastery. An unsuccessful attempt loads a different five-form bank.

QUICK ANSWERS

Surface Area and Volume FAQ

Does an open box have less volume than a closed box with the same dimensions?

No. Removing the top changes surface area, not the interior dimensions used for volume.

How are liters related to cubic centimeters?

1 L = 1,000 cm³, and 1 mL = 1 cm³.

Should I memorize every solid formula?

Know the common formulas, but connect them to base area, height, and which surfaces are present.

RELATED COMPETENCIES

Continue your mathematics review.

Perimeter and AreaCirclesRatio and ProportionUnit Conversion

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