PSHS NCE Solid Figures, Surface Area, and Volume Reviewer

TEACHER ABI PSHS NCE · QUANTITATIVE ABILITY

Solid Figures, Surface Area, and Volume

Recognize 3D shapes, count faces, edges, and vertices, and solve surface-area and volume problems without mixing them up.

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Solid Figures, Surface Area, and Volume

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Surface area is outside. Volume is inside.

Surface area = total area of all outside faces.
Volume = space inside a solid.

Cube: 6 square faces, 12 edges, 8 vertices.
Rectangular prism: 6 rectangular faces, 12 edges, 8 vertices.

VISUAL GUIDE

See the solid first, then the net

First look at a simple 3D drawing with only the outside edges. Then look at the net. If the solid feels hard to picture, the net makes the faces easier to count and is often easier for surface area.

CUBE

3D view

Net view

If the cube sketch feels tricky, the net shows the 6 square faces more clearly.

Cube formulas: Volume = s³ · Surface area = 6s²

RECTANGULAR PRISM

3D view

Net view

The prism’s net helps you see all 6 rectangular faces, which is useful when finding surface area.

Rectangular prism formulas: Volume = L × W × H · Surface area = 2(LW + LH + WH)

TRIANGULAR PRISM

3D view

Net view

In the 3D view, it can be hard to keep track of the side faces. The net makes it easier: 3 rectangles and 2 triangles.

Triangular prism volume: area of triangular base × prism length

SQUARE PYRAMID

3D view

Net view

The pyramid sketch shows the shape, but the net makes the faces easiest to count: 1 square and 4 triangles.

Square pyramid volume: 1/3 × base area × height

WORKED EXAMPLES

Six patterns you should recognize

1. Count parts of a cube

A cube has 6 faces, 12 edges, and 8 vertices.

2. Rectangular prism volume

L=8 cm, W=5 cm, H=3 cm.

8 × 5 × 3 = 120 cm³

120 cm³

3. Cube surface area

Side length = 4 cm.

6 × 4² = 6 × 16 = 96 cm²

96 cm²

4. Rectangular prism surface area

L=6, W=4, H=2.

2(24 + 12 + 8) = 88

88 square units

5. Triangular prism volume

Triangular base: b=6 cm, h=4 cm. Prism length=10 cm.

(1/2 × 6 × 4) × 10 = 120 cm³

120 cm³

6. Square pyramid volume

Square base side=6 cm, height=9 cm.

1/3 × 36 × 9 = 108 cm³

108 cm³

TRY WITH ME

Outside or inside?

A gift box is 10 cm long, 6 cm wide, and 4 cm high. How much space is inside the box?

1. Space inside means volume.
2. Use L × W × H.
3. 10 × 6 × 4 = 240.

240 cm³

COMMON TRAPS

Watch out for these

Using square units for volume.
Volume uses cubic units, such as cm³.

Using cubic units for surface area.
Surface area uses square units, such as cm².

Counting hidden edges twice.
Count each edge once, even if it is dashed or hidden.

Forgetting the 1/3 for pyramids.
A pyramid with the same base and height as a prism has one third the volume.

REGULAR PRACTICE

Build the skill

QUICK FIRE

Can you tell the solid fact fast?

Question 1 of 5

FRESH MASTERY CHECK

Can you do it without hints?

Score 5/5 to verify this competency.

NEED HELP?

Still mixing up surface area and volume?

Send Teacher Abi the item that confused you. I can help you identify whether the problem is asking about the outside or the inside.

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PSHS NCE REVIEW PATH

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Next competency: Measurement and Unit Conversion

Previous: Perimeter and AreaPSHS NCE Student Hub

PSHS NCE Perimeter and Area Reviewer

TEACHER ABI PSHS NCE · QUANTITATIVE ABILITY

Perimeter and Area

Know whether a problem is asking for the boundary or the space inside, then use the shortest correct formula.

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Around or inside?

Perimeter = distance around the outside.
Area = amount of flat space inside.

Fast clue: fence, border, frame, edge, around → think perimeter.
floor, field, tile, cover, inside → think area.

VISUAL GUIDE

See what each formula measures

RECTANGLE

Rectangle: P = 2L + 2W, A = L × W

TRIANGLE

Triangle area: A = 1/2 × base × height

PARALLELOGRAM

Parallelogram area: A = base × perpendicular height

TRAPEZOID

Trapezoid area: A = 1/2 × (b₁ + b₂) × h

WORKED EXAMPLES

Six patterns you should recognize

1. Rectangle perimeter

Length 9 cm, width 4 cm.

2(9) + 2(4) = 26 cm

26 cm

2. Rectangle area

Length 9 cm, width 4 cm.

9 × 4 = 36 cm²

36 cm²

3. Square

A square has side 7 m.

P = 4(7) = 28 m
A = 7 × 7 = 49 m²

4. Triangle area

Base 12 cm, height 5 cm.

1/2 × 12 × 5 = 30 cm²

30 cm²

5. Parallelogram area

Base 10 m, perpendicular height 6 m.

10 × 6 = 60 m²

60 m²

6. Missing dimension

A rectangle has area 48 cm² and width 6 cm. Find the length.

48 ÷ 6 = 8 cm

8 cm

TRY WITH ME

Choose perimeter or area first

A rectangular playground is 15 m long and 8 m wide. A fence will go all the way around it.

1. Fence means around the outside.
2. So use perimeter.
3. 2(15) + 2(8) = 30 + 16.

46 m of fencing

COMMON TRAPS

Watch out for these

Using perimeter for flooring.
Flooring covers inside space, so use area.

Forgetting the 1/2 in triangle area.
A triangle is half of a matching parallelogram or rectangle.

Using the slanted side as height.
Height must be perpendicular to the base.

Wrong units.
Perimeter uses linear units. Area uses square units.

REGULAR PRACTICE

Build the skill

These mix rectangle, square, triangle, parallelogram, trapezoid, and missing-dimension problems.

QUICK FIRE

Around or inside?

Five fast questions. Identify the quantity before calculating.

Question 1 of 5

FRESH MASTERY CHECK

Can you do it without hints?

Score 5/5 to verify this competency. If you miss one, a fresh set will be ready.

NEED HELP?

Still mixing up perimeter and area?

Send Teacher Abi the exact item that confused you. I can help you identify what the problem is measuring before you calculate.

Message Teacher Abi

PSHS NCE REVIEW PATH

Choose your next step

Next competency: Solid Figures, Surface Area, and Volume

Previous: Angles and Basic GeometryPSHS NCE Student Hub

PSHS NCE Angles and Basic Geometry Reviewer

TEACHER ABI PSHS NCE · QUANTITATIVE ABILITY

Angles and Basic Geometry

See the shapes, not just the words. This reviewer uses simple diagrams to teach angle relationships and basic plane-geometry ideas in a Grade 6–7 friendly way.

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Angles and Basic Geometry

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Basic ideas to keep in mind

Point = exact location
Line segment = has two endpoints
Ray = starts at one point and goes on in one direction
Angle = formed by two rays with the same endpoint

Quick facts:
Right angle = 90°
Straight angle = 180°
Full turn = 360°
Angles in a triangle = 180°

VISUAL GUIDE

See the relationships

COMPLEMENTARY ANGLES

Complementary angles add up to 90°.

SUPPLEMENTARY ANGLES

Supplementary angles add up to 180°.

VERTICAL ANGLES

Vertical angles are equal.

TRIANGLE ANGLE SUM

The interior angles of a triangle total 180°.

WORKED EXAMPLES

Six patterns you should recognize

1. Identify angle types

An angle measuring 35° is acute. An angle measuring 90° is a right angle. An angle measuring 135° is obtuse.

2. Complementary angles

x + 37 = 90

Subtract 37 from 90.

x = 53°

3. Supplementary angles

x + 125 = 180

Subtract 125 from 180.

x = 55°

4. Vertical angles

If one angle in an X-shaped intersection is 72°, the vertical angle across from it is also 72°.

Vertical angles are equal.

5. Angles in a triangle

48 + 67 + x = 180

Add 48 and 67 to get 115. Then 180 − 115 = 65.

x = 65°

6. Parallel and perpendicular lines

Parallel lines never meet. Perpendicular lines meet to form a right angle.

Shortcut: If the corner is 90°, the lines are perpendicular.

TRY WITH ME

Which relationship is it?

Two angles form a straight line. One angle is 146°. What is the other angle?

1. A straight line means the two angles are supplementary.
2. Supplementary angles add to 180°.
3. 180 − 146 = 34.

34°

COMMON TRAPS

Watch out for these

Mixing up complementary and supplementary.
Complementary = 90°. Supplementary = 180°.

Forgetting triangle angle sum.
Angles inside a triangle always total 180°.

Not recognizing vertical angles.
Opposite angles in an X-shape are equal.

Using a picture only by appearance.
Use the given measurements and facts, not just how “big” the drawing looks.

REGULAR PRACTICE

Build the skill

These mix angle types, complementary, supplementary, vertical angles, triangle angles, and line relationships.

QUICK FIRE

Can you spot the angle fact fast?

Five quick questions. Use the shortest valid idea.

Question 1 of 5

FRESH MASTERY CHECK

Can you do it without hints?

Score 5/5 to verify this competency. If you miss one, a fresh set will be ready.

NEED HELP?

Still mixing up angle relationships?

Send Teacher Abi your result or the exact item that confused you. I can help you see which geometry fact to use first.

Message Teacher Abi

PSHS NCE REVIEW PATH

Choose your next step

Next competency: Perimeter and Area

Previous: Rates, Work, and Distance PSHS NCE Student Hub

PSHS NCE Geometry and Measurement Reviewer

TEACHER ABI PSHS NCE · QUANTITATIVE ABILITY

Geometry and Measurement

Use perimeter, area, angle facts, and unit conversions quickly and accurately.

Grade 6–7 friendlyPerimeter + AreaAngles + UnitsQuick Fire + Mastery

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Know what the question is measuring

Perimeter is the distance around a shape.

Area is the amount of space inside a flat shape.

Rectangle:
Perimeter = 2L + 2W
Area = L × W

DO IT FAST

Use IDENTIFY → FORMULA → UNIT

1. IDENTIFY: Is the problem asking for perimeter, area, angle, or measurement?
2. FORMULA: Use only the formula you need.
3. UNIT: Check cm vs cm², m vs m², minutes vs hours, and so on.
4. REASON: Ask whether your answer is the right size.

WORKED EXAMPLES

Six patterns you should recognize

1. Rectangle perimeter

Length 8 cm, width 5 cm.

2(8) + 2(5) = 26 cm

26 cm

2. Rectangle area

Length 9 m, width 4 m.

9 × 4 = 36 m²

36 m²

3. Triangle area

Base 10 cm, height 6 cm.

1/2 × 10 × 6 = 30 cm²

30 cm²

4. Angle fact

Angles on a straight line add to 180°. If one angle is 125°, the other is:

180 − 125 = 55°

55°

5. Unit conversion

3.5 m = ? cm

3.5 × 100 = 350 cm

350 cm

6. Time conversion

2.5 hours = ? minutes

2.5 × 60 = 150 minutes

150 minutes

TRY WITH ME

Perimeter or area?

A rectangular garden is 12 m long and 7 m wide. How much fencing is needed to go around it?

1. Fencing goes around the garden.
2. So we need perimeter.
3. 2(12)+2(7)=24+14.

38 m

COMMON TRAPS

Watch out for these

Using area when the question asks “around.”
Around means perimeter.

Forgetting square units.
Area uses cm², m², and so on.

Using a slanted side as triangle height.
The height must be perpendicular to the base.

Converting units in the wrong direction.
1 m = 100 cm, so meters to centimeters gets larger.

REGULAR PRACTICE

Build the skill

These mix perimeter, area, angles, and measurement conversions.

QUICK FIRE

Can you choose the right formula fast?

Five short questions. Decide what is being measured before calculating.

Question 1 of 5

FRESH MASTERY CHECK

Can you do it without hints?

Score 5/5 to verify this competency. If you miss one, a fresh set will be ready.

NEED HELP?

Geometry formulas getting mixed up?

Send Teacher Abi the problem that confused you and your result. I can help you identify what the question is actually measuring.

Message Teacher Abi

PSHS NCE REVIEW PATH

Choose your next step

Next competency: Patterns, Sequences, and Number Relationships

Previous: Rates, Work, and DistancePSHS NCE Student Hub

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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