Surface Area and Volume

📦 Surface Area and Volume

Part 1 of 5 — Solids, Nets, and Two Big Ideas

Topics in This Part

🔑 Key Concept: A flat shape (like a rectangle) is 2D. A solid you can hold (like a box) is 3D. This whole lesson is about measuring two things on solids: the skin that wraps around them (surface area) and the space inside them (volume).

The Parts of a Solid

Every solid (a 3D shape) is built from three kinds of pieces:

• A face is a flat surface — like one side of a box.
• An edge is a line where two faces meet.
• A vertex is a corner point where edges meet. (More than one = vertices.)

A Rectangular Prism (a Box)

A rectangular prism — the shape of a cereal box or a brick — has:

💡 The faces of a box come in matching pairs: top matches bottom, front matches back, left matches right. That "pairs" idea makes surface area much easier later in Part 3.

Concept Check 🎯

Surface Area vs. Volume

These two ideas sound similar but measure completely different things.

Surface area adds up the area of every face — it is just a bunch of 2D areas put together, so it stays in square units like $\text{cm}^2$.

Volume counts how many unit cubes fit inside, so it uses cubic units like $\text{cm}^3$.

⚠️ The #1 mix-up: square units ($\text{cm}^2$) are for covering a surface; cubic units ($\text{cm}^3$) are for filling a space. Always ask: am I covering the outside, or filling the inside?

Pick the Right Unit 🔽

For each job, choose what you are really measuring and the correct kind of unit.

Unfolding a Solid: The Net

A net is what you get when you "unfold" a solid and lay all of its faces out flat — like cutting open a cardboard box and flattening it.

A net is powerful because it turns a scary 3D problem into easy 2D rectangles:

• The area of the net equals the surface area of the solid.
• You can find each flat face's area separately, then add them all up.

Example: the net of a cube

A cube has $6$ identical square faces. Its net is $6$ equal squares joined together. If you find the area of just one square and there are $6$ of them, the surface area is six times that single area.

🔑 Key Idea: Surface area = the area of the net = the sum of the areas of all the faces. We will use this exact strategy in Parts 3 and 4.

Concept Check 🎯

📦 Surface Area and Volume

Part 2 of 5 — Volume of Rectangular Prisms

🔑 The Idea: Volume counts how many $1\times1\times1$ unit cubes fit inside a solid. For a box, we can count them with one short formula instead of stacking cubes one by one.

Counting Unit Cubes

A unit cube is a tiny cube that is $1$ unit on every side. Its volume is $1$ cubic unit.

Suppose a box is $4$ units long, $3$ units wide, and $2$ units tall.

1. The bottom layer holds $4 \times 3 = 12$ cubes.
2. The box is $2$ layers tall, so we stack $2$ of those layers.
3. Total cubes: $12 \times 2 = 24$.

$V = 4 \times 3 \times 2 = 24 \text{ cubic units}$

💡 "One layer, then stack the layers" is the whole idea behind the volume formula. The bottom layer is the area of the base, and the height tells you how many layers to stack.

The Volume Formula

For a rectangular prism:

$V = l \times w \times h$

where $l$ is length, $w$ is width, and $h$ is height.

Because the bottom layer is the area of the base, $B = l \times w$, we can also write the formula that works for every prism:

$V = B \times h \quad (\text{base area} \times \text{height})$

Worked Example

A box is $5\text{ cm}$ long, $4\text{ cm}$ wide, and $3\text{ cm}$ tall.

$V = l \times w \times h = 5 \times 4 \times 3 = 60 \text{ cm}^3$

⚠️ Volume always uses cubic units (a small $3$, like $\text{cm}^3$) because you multiplied three lengths together.

Concept Check 🎯

A Tip Before You Compute

When three numbers are multiplied, you can do it in any order — pick the easiest pair first.

For $V = 4 \times 5 \times 2$, the pair $5 \times 2 = 10$ is friendly, so $10 \times 4 = 40$. Same answer, less effort.

💡 Look for pairs that make $10$, $100$, or another round number. It keeps your arithmetic clean on the drill below.

Compute the Volume 🧮

Enter the volume as a number only (the unit is already shown). Use $V = l \times w \times h$.

1) $l = 7\text{ cm},\ w = 3\text{ cm},\ h = 2\text{ cm}\ \Rightarrow\ V = \,?\ \text{cm}^3$ 2) A cube with every edge $4\text{ cm}\ \Rightarrow\ V = \,?\ \text{cm}^3$ 3) $l = 10\text{ ft},\ w = 5\text{ ft},\ h = 6\text{ ft}\ \Rightarrow\ V = \,?\ \text{ft}^3$

Edge Lengths That Are Fractions

The formula $V = l \times w \times h$ still works even when the sides are fractions or decimals — you just multiply carefully.

Worked Example (fraction)

A box is $\dfrac{1}{2}\text{ ft}$ long, $\dfrac{1}{2}\text{ ft}$ wide, and $3\text{ ft}$ tall.

$V = \frac{1}{2} \times \frac{1}{2} \times 3 = \frac{1}{4} \times 3 = \frac{3}{4} \text{ ft}^3$

Worked Example (decimal)

A box is $2.5\text{ cm}$ by $2\text{ cm}$ by $4\text{ cm}$.

$V = 2.5 \times 2 \times 4 = 5 \times 4 = 20 \text{ cm}^3$

💡 To multiply fractions, multiply the tops together and the bottoms together: $\dfrac{1}{2}\times\dfrac{1}{2}=\dfrac{1\times 1}{2\times 2}=\dfrac{1}{4}$.

Volume with Fractions & Decimals 🧮

Enter each answer as a number only. Fractions like $3/4$ and decimals like $0.75$ are both fine.

1) $l = \dfrac{1}{2}\text{ cm},\ w = 4\text{ cm},\ h = 3\text{ cm}\ \Rightarrow\ V = \,?\ \text{cm}^3$ 2) $l = 1.5\text{ m},\ w = 2\text{ m},\ h = 2\text{ m}\ \Rightarrow\ V = \,?\ \text{m}^3$

📦 Surface Area and Volume

Part 3 of 5 — Surface Area of Boxes & Cubes

🔑 The Idea: Surface area is the area of every face added together. For a box, the faces come in $3$ matching pairs, so we find $3$ areas, add them, and double the total.

Add Up the Faces

A rectangular prism has $6$ faces in $3$ matching pairs:

• Top and bottom, each $l \times w$
• Front and back, each $l \times h$
• Left and right, each $w \times h$

So the surface area is:

$SA = 2(l \times w) + 2(l \times h) + 2(w \times h)$

A neat shortcut is to find the $3$ different face areas, add them, and then double:

$SA = 2 \big[ (l \times w) + (l \times h) + (w \times h) \big]$

⚠️ Surface area uses square units ($\text{cm}^2$), not cubic — you are adding up flat areas, not filling space.

Worked Example

Find the surface area of a box that is $l = 5\text{ cm}$, $w = 3\text{ cm}$, $h = 2\text{ cm}$.

Step 1 — the three different face areas:

$l \times w = 5 \times 3 = 15$ $l \times h = 5 \times 2 = 10$ $w \times h = 3 \times 2 = 6$

Step 2 — add them:

$15 + 10 + 6 = 31$

Step 3 — double (each area appears twice):

$SA = 2 \times 31 = 62 \text{ cm}^2$

✅ Check by counting all six faces: $15 + 15 + 10 + 10 + 6 + 6 = 62\text{ cm}^2$ ✓

Concept Check 🎯

Surface Area of a Cube

A cube has $6$ identical square faces, each with area $s \times s = s^2$. So:

$SA = 6 \times s^2 = 6s^2$

Worked Example

A cube has edges of $4\text{ cm}$.

$SA = 6 \times s^2 = 6 \times (4 \times 4) = 6 \times 16 = 96 \text{ cm}^2$

🔑 Key Idea: For a cube, find the area of one face ($s^2$), then multiply by $6$. That's it.

Build the Surface Area 🔽

A box has $l = 6\text{ cm}$, $w = 4\text{ cm}$, $h = 2\text{ cm}$. Fill in each step.

Find the Surface Area 🧮

Enter each answer as a number only (square units).

1) A box with $l = 5\text{ cm}$, $w = 4\text{ cm}$, $h = 3\text{ cm}\ \Rightarrow\ SA = \,?\ \text{cm}^2$ 2) A cube with edge $3\text{ cm}\ \Rightarrow\ SA = \,?\ \text{cm}^2$

📦 Surface Area and Volume

Part 4 of 5 — Triangular Prisms

🔑 The Idea: A triangular prism is like a box, but its two ends are triangles instead of rectangles (think of a tent or a Toblerone bar). The same two ideas — base area times height for volume, and net for surface area — still work.

Volume of a Triangular Prism

For any prism, $V = B \times h$, where $B$ is the area of the base and $h$ is how long the prism is.

For a triangular prism, the base is a triangle, so:

$B = \frac{1}{2} \times b \times h_{\triangle}$

Here $b$ is the triangle's base and $h_{\triangle}$ is the triangle's height. Then multiply by the prism's length $\ell$:

$V = B \times \ell = \left( \frac{1}{2} \times b \times h_{\triangle} \right) \times \ell$

Worked Example

A triangular prism has a triangular end with base $6\text{ cm}$ and height $4\text{ cm}$, and the prism is $10\text{ cm}$ long.

$B = \frac{1}{2} \times 6 \times 4 = \frac{1}{2} \times 24 = 12 \text{ cm}^2$ $V = B \times \ell = 12 \times 10 = 120 \text{ cm}^3$

💡 The triangle area $\frac{1}{2} \times b \times h$ is exactly half of the rectangle $b \times h$ — a triangle fills half its surrounding rectangle.

Concept Check 🎯

Two Steps, Every Time

Volume of a triangular prism is always the same two-step move:

1. Find the triangle's area first: $B = \frac{1}{2} \times b \times h_{\triangle}$.
2. Multiply by the prism's length: $V = B \times \ell$.

⚠️ Don't multiply all three numbers straight across! You must take half for the triangle before multiplying by the length, or your volume will be twice too big.

Volume of a Triangular Prism 🧮

Enter each answer as a number only (cubic units). First find the triangle's area $B = \frac{1}{2} b h_{\triangle}$, then $V = B \times \ell$.

1) Triangle base $4\text{ cm}$, triangle height $3\text{ cm}$, prism length $10\text{ cm}$. $V = \,?\ \text{cm}^3$ 2) Triangle base $10\text{ m}$, triangle height $6\text{ m}$, prism length $2\text{ m}$. $V = \,?\ \text{m}^3$

Surface Area of a Triangular Prism

Use the net. A triangular prism unfolds into $5$ faces:

• $2$ triangles (the two ends), each $\frac{1}{2} \times b \times h_{\triangle}$
• $3$ rectangles (the sides), one for each side of the triangle, each rectangle being a side-length times the prism length $\ell$

Just find the area of all $5$ faces and add them up.

Worked Example

A triangular prism has a triangle with base $6\text{ cm}$, height $4\text{ cm}$, the three triangle sides $6\text{ cm}$, $5\text{ cm}$, $5\text{ cm}$, and prism length $10\text{ cm}$.

Two triangles: each $= \frac{1}{2} \times 6 \times 4 = 12$, so together $2 \times 12 = 24\text{ cm}^2$.

Three rectangles (side $\times$ length $10$): $6 \times 10 = 60, \quad 5 \times 10 = 50, \quad 5 \times 10 = 50$

Add everything: $SA = 24 + 60 + 50 + 50 = 184 \text{ cm}^2$

🔑 Key Idea: Surface area of any prism = (the two ends) + (the rectangles that wrap around the sides). Counting faces from the net never lets you down.

Surface Area from the Net 🔽

Use the worked example above (triangle base $6$, height $4$, sides $6, 5, 5$; prism length $10$).

Pulling the Net Together

You just found every piece of the net for that prism:

All that's left is to add them up to get the total surface area. Do it in the drill below.

Total Surface Area 🧮

Finish the worked example. The two triangles total $24\text{ cm}^2$ and the three rectangles are $60$, $50$, and $50\text{ cm}^2$.

1) Add all $5$ faces: $SA = \,?\ \text{cm}^2$

📦 Surface Area and Volume

Part 5 of 5 — Real-World Problems & Mastery Check

You can now (1) name the parts of a solid, (2) find the volume of prisms, (3) find the surface area of boxes, cubes, and triangular prisms, and (4) choose the right kind of unit. Let's put it together in real situations.

Quick Reference

⚠️ Decide first: filling the inside → volume (cubic units). Covering the outside → surface area (square units).

Which Measurement? 🔽

For each story, choose whether you need surface area or volume.

A Real-World Worked Example

A toy chest is a rectangular prism: $l = 4\text{ ft}$, $w = 2\text{ ft}$, $h = 3\text{ ft}$.

How much space is inside (volume)? $V = l \times w \times h = 4 \times 2 \times 3 = 24 \text{ ft}^3$

How much wood covers the outside (surface area)? $lw = 4\times2 = 8, \quad lh = 4\times3 = 12, \quad wh = 2\times3 = 6$ $SA = 2(8 + 12 + 6) = 2 \times 26 = 52 \text{ ft}^2$

💡 Same chest, two very different numbers — because volume ($24\text{ ft}^3$) fills the inside while surface area ($52\text{ ft}^2$) wraps the outside.

Real-World Practice 🧮

A fish tank is a rectangular prism: $l = 8\text{ in}$, $w = 5\text{ in}$, $h = 4\text{ in}$. Enter each answer as a number only.

1) How much water fills the tank? $V = \,?\ \text{in}^3$ 2) How much glass covers the whole outside? $SA = \,?\ \text{in}^2$

Mixed Practice 🎯

Exit Quiz ✅

Answer all three to finish the lesson.