The Lumenary · Grades 3–5

Area, perimeter and angles

Name: ______________________

Date: ______________________

You will learn to

Find areas by counting unit squares and by multiplying length by width, and find perimeters by adding the lengths of the sides
Show that shapes with the same perimeter can have different areas, and shapes with the same area can have different perimeters
Measure and draw angles in degrees as amounts of turn, and sort triangles and quadrilaterals by their sides and angles
Explain why the angles of a triangle add up to 180°
Find the volume of a rectangular box by counting unit cubes

1.
A rectangular rug is 7 m long and 4 m wide. What is its area, in square metres?
2.
A garden is a 6 m by 4 m rectangle with a 2 m by 2 m square cut out of one corner (for a shed). What is the area of the garden, in square metres?
3.
The same garden: a 6 m by 4 m rectangle with a 2 m by 2 m square cut out of one corner. What is its perimeter, in metres? (Careful — it may surprise you!)
4.
Kofi has 20 m of fence for a rectangular pen. Which pen gives his goats the most grass?
- A.1 m by 9 m
- B.2 m by 8 m
- C.4 m by 6 m
- D.5 m by 5 m
5.
Two angles of a triangle are 47° and 68°. How many degrees is the third angle?
6.
A triangle has angles 90°, 45° and 45°. Which words describe it? Select all that apply.
- A.right
- B.isosceles
- C.acute
- D.equilateral
- E.obtuse
7.
Musa draws two angles. Both open exactly as wide as the corner of a book. The first has arms 2 cm long; the second has arms 10 cm long. Which angle is bigger?
- A.The first one
- B.The second one, because its arms are longer
- C.They are the same size: both are 90°
8.
A box is 5 cm long, 3 cm wide and 4 cm high. How many 1 cm cubes fit exactly inside it? (This is its volume in $\text{cm}^3$ .)
9.
In Lean, triangle_angle_sum has the hypotheses straight : d + c + e = 180, alt₁ : d = a and alt₂ : e = b, and concludes a + b + c = 180. Which sentence describes what Lean has checked?
- A.Lean checked that if the straight-line fact and the two alternate-angle facts hold, then the angles add up to 180.
- B.Lean checked that alternate angles at parallel lines are always equal.
- C.Lean measured many triangles and found angle sums of 180.
- D.Lean checked the rule for one particular triangle.
10.
Complete the two functions so that area(w, h) returns the area and perimeter(w, h) returns the perimeter of a rectangle w units wide and h units high. The tests then check Kofi's pens.
Computational — Practice

Answer key — Area, perimeter and angles

1.
The rug is covered by 4 rows of 7 unit squares, so its area is $4 \times 7 =$ 28 $\text{m}^2$ .
2.
The whole rectangle has area $6 \times 4 = 24\text{ m}^2$ . The shed square has area $2 \times 2 = 4\text{ m}^2$ . So the garden has area $24 - 4 =$ 20 $\text{m}^2$ . Check by splitting: a 6 m by 2 m strip ( $12\text{ m}^2$ ) and a 4 m by 2 m strip ( $8\text{ m}^2$ ) make $20\text{ m}^2$ .
3.
Walking round the six sides: $6 + 2 + 2 + 2 + 4 + 4 = 20$ m. (Bottom 6, right side up 2, inward 2, up 2, top 4, left side 4.) The perimeter is 20 m — exactly the same as the whole 6 by 4 rectangle, $2 \times (6 + 4) = 20$ ! The two new edges of the cut have the same lengths as the two pieces of edge they replace. So cutting out the corner made the area smaller but left the perimeter the same.
4.
All four pens have perimeter 20 m. Their areas are $9$ , $16$ , $24$ and $25\text{ m}^2$ . The 5 m by 5 m square holds the most. Same perimeter, different areas.
5.
The angles add up to $180^\circ$ , and $47 + 68 = 115$ . So the third angle is $180 - 115 =$ 65°.
6.
It has a $90^\circ$ angle, so it is a right triangle. Its other two angles are equal ( $45^\circ$ each), so two of its sides are equal: it is also isosceles. It is the shape you get by cutting a square in half along a diagonal.
7.
Both angles open as wide as the corner of a book, so both are right angles: $90^\circ$ . The size of an angle is the amount of turn between its arms; drawing the arms longer does not change it. They are the same size.
8.
One layer on the bottom holds $5 \times 3 = 15$ cubes. There are 4 layers, so the box holds $15 \times 4 =$ 60 cubes. Its volume is $60\text{ cm}^3$ .
9.
A theorem with hypotheses says: if the hypotheses are true, then the conclusion is true. Here the hypotheses are the two geometric facts from the human proof (angles along a straight line make 180°, alternate angles are equal), and Lean checks that $a + b + c = 180$ follows from them, for all numbers $a, b, c, d, e$ . Lean has not proved the facts about parallel lines in this file — those are assumed.
10.
```
def area(w, h):
    return w * h


def perimeter(w, h):
    return 2 * (w + h)
```
The tests try every pen with perimeter 24 and check that the $6 \times 6$ square has the largest area, $36$ .