Geometry & Statistics

Area of Polygons, Nets, and Surface Area

Grade 6

✓By the end of this lesson students will be able to calculate the area of triangles, rectangles, and parallelograms.
✓By the end of this lesson students will be able to decompose complex polygons into simpler shapes to find their total area.
✓By the end of this lesson students will be able to represent three-dimensional figures using nets.
✓By the end of this lesson students will be able to calculate the surface area of three-dimensional figures using nets.

Key concepts

Area

Area is the amount of two-dimensional space a flat shape covers. It is measured in square units, such as square inches (in²) or square centimeters (cm²).

Polygon

A polygon is a closed two-dimensional figure made up of straight line segments.

Area of a Rectangle

The area of a rectangle is found by multiplying its length by its width.

A = l × w

Area of a Square

The area of a square is found by multiplying its side length by itself.

A = s²

Area of a Triangle

The area of a triangle is half the product of its base and its perpendicular height. The height is the perpendicular distance from the base to the opposite vertex.

A = ½ × b × h

Area of a Parallelogram

The area of a parallelogram is found by multiplying its base by its perpendicular height. The height is the perpendicular distance between the base and the opposite side.

A = b × h

Composite Figures

Composite figures are shapes made up of two or more basic geometric shapes (like rectangles, triangles, or parallelograms). To find their area, you can decompose them into these simpler shapes and add their individual areas.

Net

A net is a two-dimensional pattern that can be folded to form a three-dimensional figure. It shows all the faces of the 3D figure laid out flat.

Surface Area

Surface area is the total area of all the faces (surfaces) of a three-dimensional figure. It is measured in square units.

Prism

A prism is a three-dimensional figure with two parallel and congruent bases, and rectangular faces connecting them. Examples include rectangular prisms and triangular prisms.

Pyramid

A pyramid is a three-dimensional figure with one base and triangular faces that meet at a single point called the apex. Examples include square pyramids and triangular pyramids.

Key facts to remember

1Area is the measure of the two-dimensional space a shape covers, expressed in square units.
2The area formulas are: Rectangle (A = l × w), Square (A = s²), Triangle (A = ½ × b × h), Parallelogram (A = b × h).
3The height (h) in the triangle and parallelogram area formulas must always be perpendicular to the base (b).
4A net is a two-dimensional representation of a three-dimensional figure that can be folded to form the figure.
5Surface area is the sum of the areas of all the faces of a three-dimensional figure.
6To find the area of a composite figure, decompose it into simpler polygons and add their individual areas.

Worked examples

Example 1

Find the area of the polygon shown below. The figure is composed of a rectangle with a triangle on top. The rectangle has a length of 8 inches and a width of 5 inches. The triangle has a base of 8 inches and a height of 3 inches.

IStep 1: Decompose the polygon into simpler shapes. This polygon can be seen as a rectangle and a triangle.

IIStep 2: Calculate the area of the rectangle. Formula: A = l × w. A_rectangle = 8 inches × 5 inches = 40 square inches.

IIIStep 3: Calculate the area of the triangle. Formula: A = ½ × b × h. A_triangle = ½ × 8 inches × 3 inches = ½ × 24 square inches = 12 square inches.

IVStep 4: Add the areas of the rectangle and the triangle to find the total area. Total Area = A_rectangle + A_triangle = 40 in² + 12 in² = 52 in².

Answer

52 in²

Always remember to use the perpendicular height for triangles and parallelograms.

Example 2

Draw a net for a rectangular prism with a length of 6 cm, a width of 4 cm, and a height of 3 cm. Then, calculate its surface area.

IStep 1: Draw the net. A rectangular prism has 6 faces: a top, a bottom, a front, a back, a left side, and a right side. The net will show these faces unfolded.

IIStep 2: Identify the dimensions of each face.

III - Top and Bottom faces: 6 cm × 4 cm

IV - Front and Back faces: 6 cm × 3 cm

V - Left and Right faces: 4 cm × 3 cm

VIStep 3: Calculate the area of each unique face.

VII - Area of Top/Bottom = 6 cm × 4 cm = 24 cm²

VIII - Area of Front/Back = 6 cm × 3 cm = 18 cm²

9 - Area of Left/Right = 4 cm × 3 cm = 12 cm²

10Step 4: Sum the areas of all six faces. Since there are two of each unique face:

11 Surface Area = 2 × (Area of Top/Bottom) + 2 × (Area of Front/Back) + 2 × (Area of Left/Right)

12 Surface Area = 2 × (24 cm²) + 2 × (18 cm²) + 2 × (12 cm²)

13 Surface Area = 48 cm² + 36 cm² + 24 cm²

14 Surface Area = 108 cm²

Answer

108 cm²

When drawing a net, ensure all faces are connected and can be folded to form the 3D figure.

Example 3

A triangular prism has bases that are right triangles with legs of 3 feet and 4 feet, and a hypotenuse of 5 feet. The height of the prism (distance between the triangular bases) is 7 feet. Find the surface area of this triangular prism.

IStep 1: Identify the faces of the triangular prism. It has two triangular bases and three rectangular lateral faces.

IIStep 2: Calculate the area of one triangular base. For a right triangle, the legs can be considered the base and height. Formula: A = ½ × b × h.

III A_triangle = ½ × 3 ft × 4 ft = ½ × 12 ft² = 6 ft².

IVStep 3: Calculate the area of each rectangular lateral face. The dimensions of these rectangles are the height of the prism (7 ft) and each side of the triangular base (3 ft, 4 ft, 5 ft).

V - Rectangle 1 (using 3 ft leg): Area = 3 ft × 7 ft = 21 ft².

VI - Rectangle 2 (using 4 ft leg): Area = 4 ft × 7 ft = 28 ft².

VII - Rectangle 3 (using 5 ft hypotenuse): Area = 5 ft × 7 ft = 35 ft².

VIIIStep 4: Sum the areas of all five faces (two triangles and three rectangles).

9 Surface Area = 2 × (Area of one triangle) + (Area of Rectangle 1) + (Area of Rectangle 2) + (Area of Rectangle 3)

10 Surface Area = 2 × (6 ft²) + 21 ft² + 28 ft² + 35 ft²

11 Surface Area = 12 ft² + 21 ft² + 28 ft² + 35 ft²

12 Surface Area = 96 ft²

Answer

96 ft²

Make sure to account for all faces. A triangular prism has 5 faces in total: 2 triangles and 3 rectangles.

Common mistakes

✗Confusing area with perimeter; perimeter is the distance around a shape, while area is the space it covers.
✗Using incorrect units for area (e.g., linear units like 'cm' instead of square units like 'cm²').
✗Not identifying the correct perpendicular height for triangles and parallelograms, especially when the height is outside the shape.
✗Forgetting to include all faces when calculating surface area from a net, or miscounting the number of identical faces.
✗Incorrectly calculating the area of individual faces within a net.

Exam tips

★Always draw and label diagrams or nets to visualize the problem, especially for surface area questions.
★Break down complex polygons into simpler shapes (rectangles, triangles, parallelograms) before calculating their area.
★Clearly write down the formula you are using for each step and show your work.
★Double-check your calculations and ensure your final answer includes the correct square units.

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