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Geometry and Measurement 6-8 / Week 03 / Tuesday
2/6
Week 03 · Nets and Surface Area

Tuesday

A pyramid unfolds
// Unfold a box, add up its faces
⏱ about 20 min

Tuesday: A Pyramid Unfolds

Raven holds a paper pyramid over the box. "The clubhouse roof. A square base and four triangle sides."

She unfolds it. A square sits in the middle with a triangle on each edge, like a flower.

"What do you notice about the four triangles?" she asks.

"All the same," Rocket says. "Same base as the square's side, same height. Six and five."

"So four of ½ × 6 × 5," Raven says. "Plus the square, 6 × 6."

Nova drifts over the net. "The roof sits on the box," she says. "Does the roof need its base?"

Rocket stops. "No. The box top is already there. So only the four triangles." Nova hums. "A designer counts only the cardboard she cuts."

The net of a square pyramid

The net of a square pyramid: a 6 cm square with four triangles of height 5 cm

A square pyramid has five faces: one square base and four triangles that meet at the top point. Its net is a square with a triangle on each side.

This one has a 6 cm base and triangles 5 cm tall. Base: 6 × 6 = 36. One triangle: ½ × 6 × 5 = 15.

Surface area: 36 + 4 × 15 = 96 square centimeters.

The triangle height is measured on the flat net, from the middle of the base edge up to the point.

The method for any net

  1. Unfold the solid, on paper or in your head. Count the faces and match the pairs.
  2. Find the area of each different face: base × height for rectangles, half of that for triangles.
  3. Multiply each face area by how many times it appears.
  4. Add everything. Write the units: square centimeters or square units.
SolidFaces in the netSurface areaExample
Box (rectangular prism)6 rectangles in 3 pairs2 × (top + front + end)4 by 2 by 3: 52
Cube6 equal squares6 × side × sideside 3: 54
Square pyramid1 square and 4 trianglesbase + 4 × triangle6 base, 5 height: 96

A cube is a box with every edge the same, so its six faces are equal squares. Side 3: 6 × 3 × 3 = 54.

SURFACE AREA OF A PYRAMID
  • Read the question.
  • Tap your answer.
The net of a square pyramid: a 6 cm square with four triangles of height 5 cmWhat is the surface area in square centimeters?
The net of a square pyramid: a 4 cm square with four triangles of height 3 cmWhat is the surface area in square centimeters?
The net of a square pyramid: a 10 cm square with four triangles of height 8 cmWhat is the surface area in square centimeters?
CUBES AND BOXES
  • Read the question.
  • Tap your answer.
The net of a box 3 by 3 by 3 cm: six rectangles laid flatWhat is the surface area in square centimeters?
The net of a box 5 by 5 by 5 cm: six rectangles laid flatWhat is the surface area in square centimeters?
The net of a box 8 by 2 by 2 cm: six rectangles laid flatWhat is the surface area in square centimeters?
A pyramid roof has a 6 cm square base and triangles of height 5 cm. What is the area of its four triangles, without the base?
FIND THE SURFACE AREA OF THE PYRAMID WITH BASE 6 AND TRIANGLE HEIGHT 5
  • ?Find one triangle: ½ × 6 × 5 = 15.
  • ?Add the base: 36 + 60 = 96 square centimeters.
  • ?Multiply by four triangles: 4 × 15 = 60.
  • ?Find the square base: 6 × 6 = 36.
WHY THIS EXERCISEEvery net is the same routine: each kind of face once, times how many, then add.
StatementTrue or false?
A square pyramid has five faces.?
The four triangles of a square pyramid net are all the same size.?
6 × 3 × 3 = 54?
A roof that sits on a box needs cardboard for its base too.?
WHY THIS EXERCISESurface area counts the faces you cover, and a designer leaves out faces that are already there.
Try it
On paper, draw a 6 cm square. Add a triangle 5 cm tall to each side. Cut out the net.
Fold the triangles up until the points meet. Tape them, and you have the roof.
Safety first
Scissors only, with a grown-up nearby. Cut away from your fingers and walk, never run, with scissors.
Draw the net of a square pyramid with base 4 and triangle height 3. Label every face with its area.

Squares and triangles, folded up. Tomorrow is Build Lab: unfold a real box and measure every face.

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