Rectangles & triangles
🎯 In this topic you will
- Estimate the area of a triangle.
- Work out the area of triangles using rectangles.
🧠 Key Words
- area
Show Definitions
- area: The amount of space covered by a flat shape or surface.
🌱 What Do We Know?
We have a 1 kg bag of grass seed, and we know it covers exactly 20 square metres of ground. This is our special seed fact. The brown field we want to cover is a large space, so we need to figure out its size first.
📏 Measuring the Field
To find out how many bags we need, we first have to know the area of the brown field. If the field is a rectangle, we can measure its length and its width. We might use a long tape measure, count our footsteps, or even look at a map. Once we have those two numbers, we multiply them together to get the area in square metres.
🧮 From Field Size to Bags
Now comes the calculation. Imagine the field’s area is 100 square metres. We know one bag covers 20 square metres. To find out how many bags cover 100 square metres, we simply divide the total area by the area one bag covers.
The Magic Formula: $\text{Number of bags} = \dfrac{\text{Total area of field (in m²)}}{\text{Area covered by one bag (20 m²)}}$
🧾 Our Final Decision
Once we do the division, we get a number. If the answer is not a whole number, it means we need to buy a few extra bags to make sure we have enough seed for the whole field. For example, if we need 4.3 bags, we would buy 5 bags. It is always better to have a little bit left over than to run out!
❓ EXERCISES
1. This rectangle was made by putting two squares together.
a. What is the area of the rectangle?
b. What is the area of one of the squares?
👀 Show answer
1a. The rectangle is made of two squares side by side, so its width is $6$ cm and its length is $6 + 6 = 12$ cm. Area = length × width = $12 \times 6 = 72$ cm².
1b. Each square has side $6$ cm, so area = $6 \times 6 = 36$ cm².
2. This square was made by putting two identical rectangles together.
a. What is the area of the square?
b. What is the area of one of the rectangles?
👀 Show answer
2a. The square has side $9$ cm, so area = $9 \times 9 = 81$ cm².
2b. The square is made of two identical rectangles. One rectangle has area = $81 \div 2 = 40.5$ cm².
3. Asok took two pieces of paper.
a. What was the area of the piece of paper before it was cut?
b. What is the area of one of the smaller pieces of paper Asok made? He cut one piece of paper in half like this:
c. He cut the other piece of paper in half like this: What is the area of one of the smaller pieces of paper Asok made?
👀 Show answer
3a. The first piece is $30 \times 21$ cm, area = $30 \times 21 = 630$ cm². The second piece is $29 \times 7.1$ cm, area = $29 \times 7.1 = 205.9$ cm².
3b. The first piece ($30 \times 21$) cut in half (assuming lengthwise cut) gives two pieces each $15 \times 21$ cm. Area of one smaller piece = $15 \times 21 = 315$ cm².
3c. The second piece ($29 \times 7.1$) cut in half gives two pieces each $14.5 \times 7.1$ cm. Area = $14.5 \times 7.1 = 102.95$ cm².
4. Estimate the area of these triangles by counting the squares.
👀 Show answer
این یک متن توصیفی است و سوالی برای حل ندارد. این بخش صرفاً یک توضیح درباره یک تصویر (عکس یک فرد در اتاق) ارائه میدهد.
توجه: این قسمت برای توصیف تصویر است و نیازی به پاسخ ریاضی ندارد.
5. Selena made this pattern by overlapping tissue paper triangles.
Below are the bottom three triangles, as they look on a centimetre square grid.
a. Draw and complete a table to show the area of each triangle in the pattern.
b. What would be the area of the 7th triangle?
c. What would be the area of the 10th triangle?
d. Look at the pattern of numbers in your table. Try to describe the pattern of the areas of the triangles. Can you think of a way to always predict what the area of the next triangle will be? Can you describe the link between the number of each triangle and its area?
👀 Show answer
5a. Based on the centimetre grid pattern (each small square = $1$ cm²), the triangles likely increase by a constant amount each time. For example, if triangle 1 area = $1$ cm², triangle 2 = $3$ cm², triangle 3 = $5$ cm² (odd numbers), then table: triangle 1: $1$, 2: $3$, 3: $5$, 4: $7$, 5: $9$, 6: $11$, 7: $13$, 8: $15$, 9: $17$, 10: $19$.
5b. Area of 7th triangle = $13$ cm².
5c. Area of 10th triangle = $19$ cm².
5d. The areas form a pattern of odd numbers: $1, 3, 5, 7, \dots$. Each new triangle adds $2$ cm² to the previous area. The area of triangle number $n$ is $(2n - 1)$ cm². So to predict the next area, add $2$ to the last area, or use the formula $2n - 1$.
🧠 Think like a Mathematician
Question: What is the area of each rectangle below? Count squares to estimate the area of each triangle.
Equipment: Squared paper (centimetre grid), pencil, ruler, eraser
Method:
- Look carefully at each rectangle. Inside each one, a triangle has been drawn. The triangle reaches the full width and full height of the rectangle.
- Count the whole centimetre squares inside the first triangle. Then count the half squares and combine them to make whole squares.
- Add the whole and part squares together to estimate the triangle's area in square centimetres ($cm^2$).
- Repeat this counting method for the second and third triangles.
- Now take a fresh piece of squared paper and draw a rectangle that is exactly $8$ cm by $4$ cm.
- Inside this rectangle, draw your own triangles. Each triangle must be as wide and as tall as the rectangle—meaning its base is the full $8$ cm width and its height is the full $4$ cm height.
- Try drawing different triangles with the same base and height: for example, a right‑angled triangle, an isosceles triangle, and a slanted triangle. Estimate each area by counting squares.
- Write down all your estimates in a table and compare them.
Description and Follow‑up Questions:
📌 Show possible answers
- 1: The areas of the three given triangles are all very close—they are each about half the area of their surrounding rectangle. For example, if a rectangle is $3 \times 4 = 12$ cm², the triangle inside it is approximately $6$ cm².
- 2: All the triangles I drew inside the $8 \times 4$ rectangle have almost the same area (around $16$ cm²), even though they look different. This shows that the shape of the triangle doesn't change its area as long as the base and height stay the same.
- 3: The area of a triangle that has the same base and height as a rectangle is always exactly half the area of that rectangle. This can be written as: $\text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height}$.
❓ EXERCISES
6. These rectangles are cut in half diagonally to make two triangles.
For each diagram work out the area of the rectangle and the area of one of the triangles.
a. $4\,cm$ by $4\,cm$
b. $5\,cm$ by $3\,cm$
c. $5\,cm$ by $4\,cm$
d. $6\,cm$ by $2\,cm$
e. $7\,cm$ by $3\,cm$
f. $3\,cm$ by $4\,cm$
👀 Show answer
a. Rectangle area $= 4 \times 4 = 16\,cm^2$; triangle area $= 8\,cm^2$
b. Rectangle area $= 5 \times 3 = 15\,cm^2$; triangle area $= 7.5\,cm^2$
c. Rectangle area $= 5 \times 4 = 20\,cm^2$; triangle area $= 10\,cm^2$
d. Rectangle area $= 6 \times 2 = 12\,cm^2$; triangle area $= 6\,cm^2$
e. Rectangle area $= 7 \times 3 = 21\,cm^2$; triangle area $= 10.5\,cm^2$
f. Rectangle area $= 3 \times 4 = 12\,cm^2$; triangle area $= 6\,cm^2$
7. Work out the area of the rectangle.
The area of this triangle is $9\,cm^2$.
👀 Show answer
8. Jo makes triangular biscuits by cutting out $5\,cm$ squares of dough, then cutting them in half.
Jo wants to cover each biscuit in icing.
This $400\,g$ tub of icing covers $340\,cm^2$ of biscuit.
How many triangular biscuits will Jo be able to cover?