Area and surface area

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1. Area of parallelograms

Warm-up

Three quick questions from earlier lessons.

Discover

Cut a triangle off one end of a parallelogram and slide it to the other end. What shape do you get?

  1. A rectangle with the same base and height
  2. A rectangle as long as the slanted side

Key idea

Area of a parallelogram.

Area of a parallelogram =b⋅h= b \cdot h, where bb is a base and hh the matching height: the distance from the base to the opposite side, measured at a right angle.

Worked example

Base 8 cm, height 5 cm, slanted side 6 cm.

A=8⋅5A = 8 \cdot 5
The slanted side is not used.
4040 cm²
Multiply.

Your turn, guided

Your turn: base 7, height 4. The area.

2828
7 · 4.

Practice ★

  1. A parallelogram has a base of 11 cm, a height of 5 cm and slanted sides of 7 cm. What is its area (cm²)?
  2. A parallelogram has a base of 3 cm, a height of 7 cm and slanted sides of 8 cm. What is its area (cm²)?
  1. A parallelogram has a base of 9 cm. Which length is its height?
    1. The distance between the base and the opposite side, measured at a right angle
    2. The length of a slanted side
    3. The longest side

Watch out!

Practice ★★

  1. A parallelogram has an area of 27 cm² and a base of 3 cm. What is its height (cm)?
  2. A parallelogram has an area of 32 cm² and a base of 4 cm. What is its height (cm)?

Exit ticket

Three questions to check you have got it.

  1. A parallelogram has a base of 11 cm, a height of 5 cm and slanted sides of 7 cm. What is its area (cm²)?
  2. A parallelogram has a base of 3 cm, a height of 7 cm and slanted sides of 8 cm. What is its area (cm²)?
  3. A parallelogram has a base of 14 cm, a height of 8 cm and slanted sides of 12 cm. What is its area (cm²)?

Summary card

Area of parallelograms

A=b⋅hA = b \cdot h, with the height at a right angle to the base.

Watch out: the slanted side is not the height.

2. Area of triangles

Warm-up

Three quick questions from earlier lessons.

Discover

Two copies of the same triangle fit together into a parallelogram with base 6 and height 4. What is the area of one triangle?

  1. 12
  2. 24

Slide C along the dashed line. What happens to the area?

  1. It stays 12
  2. It changes with the slant of the sides

Key idea

A triangle is half a parallelogram.

Area of a triangle =12b⋅h= \tfrac{1}{2} b \cdot h, with hh the height to the chosen base (measured at a right angle; it may fall outside the triangle).

Worked example

Base 10 cm, height 7 cm.

A=12⋅10⋅7=35A = \tfrac{1}{2} \cdot 10 \cdot 7 = 35 cm²
Half of base × height.

Your turn, guided

Your turn: base 9, height 6. The area.

2727
½ · 54.

Practice ★

  1. A triangle has a base of 15 cm and a height of 4 cm. What is its area (cm²)?
  2. A triangle has a base of 11 cm and a height of 4 cm. What is its area (cm²)?
  1. A triangle has a base of 7 m, a height of 6 m and another side of 11 m. What is its area (m²)?

Watch out!

Practice ★★

  1. A triangle has an area of 32 cm² and a height of 8 cm. What is its base (cm)?
  2. A triangle has an area of 40 cm² and a height of 4 cm. What is its base (cm)?
  1. Boss problem. A tent is a prism. Its front is a triangle with base 8 ft and height 4 ft, and the tent is 10 ft long.Not to scale

Exit ticket

Three questions to check you have got it.

  1. A triangle has a base of 15 cm and a height of 4 cm. What is its area (cm²)?
  2. A triangle has a base of 11 cm and a height of 4 cm. What is its area (cm²)?
  3. A triangle has a base of 20 cm and a height of 9 cm. What is its area (cm²)?

Summary card

Area of triangles

A=12bhA = \tfrac{1}{2} b h.

Watch out: do not forget the ½.

3. Surface area of prisms

Warm-up

Three quick questions from earlier lessons.

Discover

To wrap a box you need to cover its outside. Which measure is that?

  1. Surface area
  2. Volume
Not to scale

Turn and tilt the box. Count its faces, the hidden ones too.

  1. 6 faces, in 3 matching pairs
  2. 4 faces: the ones on the sides

Key idea

Surface area adds the faces.

The surface area of a prism is the sum of the areas of all its faces: two bases plus the side rectangles (perimeter of the base × length). For a box: 2(lw+lh+wh)2(lw + lh + wh). Volume is base area × height, in cubic units.

Worked example

A box 5 by 3 by 2.

Not to scale

2(15+10+6)2(15 + 10 + 6)
Pairs of faces.
6262 square units
Add.

Your turn, guided

Your turn: the surface area of a cube with edge 3.

5454
6 · 9.

Practice ★

  1. Find the surface area of a box (rectangular prism) 6 cm by 9 cm by 5 cm (cm²).Not to scale
  2. Find the surface area of a box (rectangular prism) 7 cm by 3 cm by 7 cm (cm²).Not to scale
  1. Find the surface area of a cube with edge 10 cm (cm²).

Watch out!

Practice ★★

  1. A prism has right-triangle bases with sides 3, 4 and 5 cm, and length 7 cm. Find its surface area (cm²).Not to scale
  2. A prism has right-triangle bases with sides 9, 12 and 15 cm, and length 2 cm. Find its surface area (cm²).Not to scale
  1. Boss problem. A gift box is 6 in long, 4 in wide and 5 in tall.

Exit ticket

Three questions to check you have got it.

  1. Find the surface area of a box (rectangular prism) 6 cm by 9 cm by 5 cm (cm²).Not to scale
  2. Find the surface area of a box (rectangular prism) 7 cm by 3 cm by 7 cm (cm²).Not to scale
  3. Find the surface area of a box (rectangular prism) 8 cm by 2 cm by 3 cm (cm²).Not to scale

Summary card

Surface area

Add the areas of all faces (square units).

Watch out: covering is area; filling is volume.

Answers: Area and surface area

1. Area of parallelograms

A

  1. 5555
  2. 2121

B

  1. The distance between the base and the opposite side, measured at a right angle

C

  1. 99
  2. 88

D

  1. 5555
  2. 2121
  3. 112112

2. Area of triangles

A

  1. 3030
  2. 2222

B

  1. 2121

C

  1. 88
  2. 2020

D

  1. 8080

E

  1. 3030
  2. 2222
  3. 9090

3. Surface area of prisms

A

  1. 258258
  2. 182182

B

  1. 600600

C

  1. 9696
  2. 180180

D

  1. 124124

E

  1. 258258
  2. 182182
  3. 9292