seegongsik
Grade 5 (age 10-11)

grade 5 Area of Polygons

The area of a polygon can be found by turning it into a rectangle, whose area you already know. Cut the triangle off one end of a parallelogram and stick it on the other end, and you get a rectangle, so its area is base × height; two identical triangles make a parallelogram, so a triangle's area is base × height ÷ 2. Draw a diagonal across a trapezoid and it splits into two triangles, which gives its area formula. In the two pictures, change the base and the height to check the area of the rectangle made by cutting and moving, and of the triangle that is half of it.

The key idea

✂️ Cut and rearrange into a rectangle
①The rectangle is the shape whose area we can find directly, so other polygons are cut and rearranged into rectangles
②Cut the triangle off one end of a parallelogram and move it to the other end, and you get a rectangle with the same base and height
③Two identical triangles fit together into a parallelogram, so a triangle has half that area
④Split a trapezoid into two triangles along a diagonal and you get (top + bottom) × height ÷ 2. With top 4, bottom 6 and height 4, that is (4 + 6) × 4 ÷ 2 = 20

Area of a parallelogram

8
5

Area of a triangle

↔️ The same base and height
The base and height you chose above are used for the triangle below too. The dashed rectangle has exactly that base and height.
💡 A triangle is half a rectangle
①The dashed rectangle has the same base and height as the triangle
②The height dropped from the top vertex splits the rectangle in two, and on each side the part inside the triangle is the same size as the part outside
③So the triangle's area is half the rectangle's, base × height ÷ 2. With the starting values, 8 × 5 ÷ 2 = 20
④Wherever the top vertex is, the same base and height give the same area

Summary

Area of a parallelogram
Area = base × height
Moving the triangle from one end turns it into a rectangle
Area of a triangle
Area = base × height2
Half of the rectangle with the same base and height
Area of a trapezoid
Area = (top + bottom) × height2
The two triangles made by a diagonal, added together
🎯 At a glance
①Rectangle area = length × width is where every area formula starts
②Parallelogram = base × height
③Triangle = base × height ÷ 2
④Trapezoid = (top + bottom) × height ÷ 2
⑤The height is always measured at right angles to the base, not along a slanted side

Examples and Unit Test

Example 1
Find the area of a parallelogram with base 8 cm and height 5 cm.
1
The area of a parallelogram is base × height.
2
8 × 5 = 40, so the area is 40 cm².
▸ 40 cm²
This is the parallelogram in the first picture at its starting setting, where Area = 8 × 5 = 40 is written at the top.
Example 2
Find the area of a triangle with base 8 cm and height 5 cm.
1
The area of a triangle is base × height ÷ 2.
2
8 × 5 ÷ 2 = 40 ÷ 2 = 20, so the area is 20 cm².
▸ 20 cm²
It is half of 40 cm², the area of the parallelogram with the same base and height.
Unit Test
A trapezoid has a top of 4 cm, a bottom of 6 cm and a height of 4 cm. What is its area?
①20 cm²
②40 cm²
③24 cm²
④10 cm²
▸ ① 20 cm²
1
The area of a trapezoid is (top + bottom) × height ÷ 2.
2
(4 + 6) × 4 ÷ 2 = 40 ÷ 2 = 20, so it is 20 cm².
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Congruence & Symmetry
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Rectangular Solid: Surface Area & Volume

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