Solved Exercises on Surface Area of 3D Shapes in Grade 7

Master surface area calculations: cubes, rectangular prisms, cylinders, pyramids, and cones through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Cube Surface Area
Exercise 1
Find the surface area of a cube with side length 6 cm.
Visual: 📦 Cube with sides labeled 6 cm each
Definition:

Surface Area: Total area of all faces of a 3D shape

Cube: 6 identical square faces

Calculation method:
  1. A cube has 6 square faces
  2. Each face has area = side × side = s²
  3. Surface Area = 6 × s²
Formula
SA = 6s²
Substitute
SA = 6(6)²
Calculate
SA = 216 cm²
Step 1: Identify the formula

For a cube: Surface Area = 6 × (side length)²

SA = 6s² where s = 6 cm

Step 2: Substitute the value

SA = 6 × (6)²

SA = 6 × 36

Step 3: Calculate the result

SA = 216 cm²

Surface Area = 216 cm²
Final answer:

The surface area of the cube is 216 cm²

Applied rules:

• Cube Formula: SA = 6s² for a cube with side s

• Area of Square: side × side = s²

• Multiplication: 6 faces × area of each face

2 Rectangular Prism Surface Area
Exercise 2
Find the surface area of a rectangular prism with dimensions:
Length = 8 cm, Width = 5 cm, Height = 3 cm
Visual: 📦 Rectangular box with L=8, W=5, H=3 labeled
Definition:

Rectangular Prism: 6 rectangular faces with opposite faces equal

Faces: 2×(L×W) + 2×(L×H) + 2×(W×H)

Formula
SA = 2(LW + LH + WH)
Substitute
SA = 2(8×5 + 8×3 + 5×3)
Calculate
SA = 158 cm²
Step 1: Identify the formula

For a rectangular prism: SA = 2(LW + LH + WH)

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

Step 2: Calculate each face area

Top/Bottom faces: L × W = 8 × 5 = 40 cm²

Front/Back faces: L × H = 8 × 3 = 24 cm²

Side faces: W × H = 5 × 3 = 15 cm²

Step 3: Sum and multiply by 2

SA = 2(40 + 24 + 15) = 2(79) = 158 cm²

Surface Area = 158 cm²
Final answer:

The surface area of the rectangular prism is 158 cm²

Applied rules:

• Prism Formula: SA = 2(LW + LH + WH)

• Area of Rectangle: length × width

• Opposite Faces Equal: Each pair of opposite faces has the same area

3 Cylinder Surface Area
Exercise 3
Find the surface area of a cylinder with radius 4 cm and height 10 cm.
Use π ≈ 3.14
Visual: 🥤 Cylinder with r=4, h=10 labeled
Definition:

Cylinder: 2 circular bases + curved lateral surface

Formula: SA = 2πr² + 2πrh

Formula
SA = 2πr² + 2πrh
Substitute
SA = 2π(4)² + 2π(4)(10)
Calculate
SA ≈ 351.68 cm²
Step 1: Identify the formula

For a cylinder: SA = 2πr² + 2πrh

Where r = 4 cm, h = 10 cm

Step 2: Calculate base areas

Area of both bases = 2πr² = 2π(4)² = 2π(16) = 32π cm²

Step 3: Calculate lateral surface area

Lateral surface area = 2πrh = 2π(4)(10) = 80π cm²

Step 4: Add both parts

SA = 32π + 80π = 112π ≈ 112 × 3.14 = 351.68 cm²

Surface Area ≈ 351.68 cm²
Final answer:

The surface area of the cylinder is approximately 351.68 cm²

Applied rules:

• Cylinder Formula: SA = 2πr² + 2πrh

• Area of Circle: πr²

• Lateral Surface: 2πrh

Rules and methods, laws,...
SA = 6s²
Cube Surface Area
SA = 2(LW + LH + WH)
Rectangular Prism Surface Area
SA = 2πr² + 2πrh
Cylinder Surface Area
Pyramid
SA = Base Area + Lateral Area
Sum of base + triangular faces
Cone
SA = πr² + πrl
Base + lateral surface
Sphere
SA = 4πr²
Complete spherical surface
Key definitions:

Surface Area: Total area of all surfaces of a 3D shape

Net: 2D representation showing all faces of a 3D shape

Lateral Surface: Side surfaces excluding bases

Complete methodology:
  1. Identify the 3D shape and its dimensions
  2. Select the appropriate formula for surface area
  3. Calculate each face area separately if needed
  4. Sum all face areas to get total surface area
  5. Include units in your final answer
Tip 1: Count the number of faces to ensure you include all of them.
Tip 2: Draw a net if it helps visualize all faces.
Tip 3: Remember that opposite faces of rectangular prisms are equal.
Common errors: Forgetting to include all faces, miscounting faces, using wrong formulas.
Key formulas: Cube: 6s², Prism: 2(LW+LH+WH), Cylinder: 2πr²+2πrh.
Formulas to know by heart:

• Cube: SA = 6s²

• Rectangular Prism: SA = 2(LW + LH + WH)

• Cylinder: SA = 2πr² + 2πrh

• Pyramid: SA = Base Area + Lateral Area

• Cone: SA = πr² + πrl

Solution: Exercises 4 to 5
4 Square Pyramid Surface Area
Exercise 4
Find the surface area of a square pyramid with:
Base side = 6 cm, Slant height = 8 cm
Visual: △ Square pyramid with base side 6 and slant height 8
Definition:

Square Pyramid: 1 square base + 4 triangular faces

Formula: SA = Base Area + 4 × Triangle Area

Base Area
6² = 36 cm²
Triangle Area
½ × 6 × 8 = 24 cm²
Total SA
36 + 4(24) = 132 cm²
Step 1: Calculate base area

Base is a square with side 6 cm

Base Area = 6 × 6 = 36 cm²

Step 2: Calculate one triangular face area

Triangle Area = ½ × base × slant height

Triangle Area = ½ × 6 × 8 = 24 cm²

Step 3: Calculate total surface area

SA = Base Area + 4 × Triangle Area

SA = 36 + 4(24) = 36 + 96 = 132 cm²

Surface Area = 132 cm²
Final answer:

The surface area of the square pyramid is 132 cm²

Applied rules:

• Pyramid Formula: SA = Base Area + Lateral Area

• Area of Square: side²

• Area of Triangle: ½ × base × height

5 Cone Surface Area
Exercise 5
Find the surface area of a cone with:
Radius = 5 cm, Slant height = 13 cm
Use π ≈ 3.14
Visual: 🍦 Cone with r=5, l=13 labeled
Definition:

Cone: Circular base + curved lateral surface

Formula: SA = πr² + πrl

Base Area
π(5)² = 25π
Lateral Area
π(5)(13) = 65π
Total SA
25π + 65π = 90π ≈ 282.6 cm²
Step 1: Calculate base area

Base is a circle with radius 5 cm

Base Area = πr² = π(5)² = 25π cm²

Step 2: Calculate lateral surface area

Lateral Area = πrl = π(5)(13) = 65π cm²

Step 3: Calculate total surface area

SA = Base Area + Lateral Area

SA = 25π + 65π = 90π cm²

SA ≈ 90 × 3.14 = 282.6 cm²

Surface Area ≈ 282.6 cm²
Final answer:

The surface area of the cone is approximately 282.6 cm²

Applied rules:

• Cone Formula: SA = πr² + πrl

• Area of Circle: πr²

• Lateral Surface: πrl

Comprehensive Guide: Surface Area Formulas and Methods
SA = 6s²
Cube
Key definitions:

Surface Area: The total area of all the surfaces of a 3D shape measured in square units

Net: A 2D pattern that can be folded to form a 3D shape

Lateral Surface Area: The area of all vertical/side surfaces, excluding bases

Complete methodology:
  1. Identify the 3D shape (cube, prism, cylinder, etc.)
  2. Determine the dimensions (length, width, height, radius, etc.)
  3. Select the appropriate formula for surface area calculation
  4. Calculate individual face areas if needed
  5. Sum all face areas to find total surface area
  6. Express answer with correct units (cm², m², etc.)
Tip 1: Always count the number of faces before starting calculations.
Tip 2: Draw a net to visualize all faces of complex shapes.
Tip 3: Remember that opposite faces of rectangular prisms have equal areas.
Tip 4: In cylinders, calculate the area of both circular bases separately.
Common errors: Forgetting to include all faces, miscounting faces, using wrong formulas, forgetting to square dimensions.
Exam preparation: Memorize formulas, practice identifying shapes, focus on unit conversions.
Essential formulas to memorize:

• Cube: SA = 6s² (where s is side length)

• Rectangular Prism: SA = 2(LW + LH + WH)

• Cylinder: SA = 2πr² + 2πrh (bases + lateral surface)

• Square Pyramid: SA = s² + 4(½ × s × l) (base + 4 triangles)

• Cone: SA = πr² + πrl (base + lateral surface)

• Sphere: SA = 4πr² (advanced concept)

Comparison of 3D Shape Surface Areas
Exercise 6: Surface Area Comparison
Compare surface areas of shapes with same base area (25 cm²):
Cube (side = 5 cm), Cylinder (r = 2.82 cm, h = 5 cm), Square Prism (base 5×5, h = 5 cm)

Analysis: Different 3D shapes with same base area have different surface areas due to their unique structures.

  • Cube: SA = 6s² = 6(5)² = 150 cm²
  • Square Prism: SA = 2(25) + 4(5×5) = 50 + 100 = 150 cm²
  • Cylinder: SA = 2πr² + 2πrh ≈ 141.4 cm²

Questions & Answers

Question: I'm confused about when to use slant height versus regular height for pyramids and cones. Can you explain the difference?

Answer: Great question! The key difference lies in what you're calculating:

  • Regular Height (h): The vertical distance from the base to the apex, used for volume calculations
  • Slant Height (l): The distance along the face from the base edge to the apex, used for surface area

For surface area calculations:

  • Pyramids: Use slant height to find the area of triangular faces: ½ × base × slant height
  • Cones: Use slant height for lateral surface: πrl

Example: For a square pyramid with base side 6 cm and slant height 8 cm, each triangular face has area = ½ × 6 × 8 = 24 cm². Using regular height would give an incorrect result for surface area.

Question: Why does a cylinder have two different parts in its surface area formula (2πr² + 2πrh)?

Answer: The cylinder's surface area formula accounts for two distinct parts:

  • 2πr²: Represents the area of both circular bases (top and bottom)
  • 2πrh: Represents the lateral (curved) surface area

Think of it as unfolding the cylinder:

  • The two circles remain as the top and bottom
  • The curved side unfolds into a rectangle with width = circumference (2πr) and height (h)
  • So the lateral surface area = 2πr × h = 2πrh

Therefore, total surface area = 2(circle areas) + 1(rectangle area) = 2πr² + 2πrh

Question: How can I remember all these different surface area formulas? They seem too many to memorize!

Answer: Instead of memorizing formulas blindly, focus on understanding the logic behind each one:

  • Cube: 6 identical squares → SA = 6s²
  • Prism: 2 bases + 4 rectangles → SA = 2(base area) + perimeter × height
  • Cylinder: 2 circles + rectangle → SA = 2πr² + 2πrh
  • Pyramid: 1 base + triangular faces → SA = base + ½ × perimeter × slant height

Memory tricks:

  • Cubes: "Six faces" → 6s²
  • Prisms: "Two ends, four sides" → 2 ends + 4 rectangles
  • Cylinders: "Two caps, one wrapper" → 2πr² + 2πrh

Practice drawing nets of shapes to visualize all faces. The more you understand the concept, the less you need to rely on pure memorization!