Problem: Calculate the areas of squares, rectangles, and triangles, as well as the semi-perimeter and area of a triangle using Heron's formula.
####
Step 1: Area of a Square
The formula for the area of a square is:
\[
\text{Area} = \text{side}^2
\]
Given:
- Side length of the square = 5 cm
Calculation:
\[
\text{Area} = 5^2 = 25 \, \text{cm}^2
\]
Answer for the square:
\[
\boxed{25 \, \text{cm}^2}
\]
---
####
Step 2: Area of a Rectangle
The formula for the area of a rectangle is:
\[
\text{Area} = \text{length} \times \text{width}
\]
Given:
- Length = 11 m
- Width = 6 m
Calculation:
\[
\text{Area} = 11 \times 6 = 66 \, \text{m}^2
\]
Answer for the rectangle:
\[
\boxed{66 \, \text{m}^2}
\]
---
####
Step 3: Area of a Triangle (Right Triangle)
The formula for the area of a triangle is:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Given:
- Base = 10 cm
- Height = 8 cm
Calculation:
\[
\text{Area} = \frac{1}{2} \times 10 \times 8 = \frac{1}{2} \times 80 = 40 \, \text{cm}^2
\]
Answer for the right triangle:
\[
\boxed{40 \, \text{cm}^2}
\]
---
####
Step 4: Semi-perimeter and Area of a Triangle Using Heron's Formula
The formula for the semi-perimeter \( s \) is:
\[
s = \frac{a + b + c}{2}
\]
where \( a \), \( b \), and \( c \) are the sides of the triangle.
The formula for the area of a triangle using Heron's formula is:
\[
\text{Area} = \sqrt{s(s - a)(s - b)(s - c)}
\]
Given:
- \( a = 8 \, \text{cm} \)
- \( b = 7 \, \text{cm} \)
- \( c = 5 \, \text{cm} \)
#####
Step 4.1: Calculate the semi-perimeter \( s \)
\[
s = \frac{a + b + c}{2} = \frac{8 + 7 + 5}{2} = \frac{20}{2} = 10 \, \text{cm}
\]
#####
Step 4.2: Calculate the area using Heron's formula
Substitute \( s = 10 \), \( a = 8 \), \( b = 7 \), and \( c = 5 \) into the formula:
\[
\text{Area} = \sqrt{s(s - a)(s - b)(s - c)}
\]
\[
\text{Area} = \sqrt{10(10 - 8)(10 - 7)(10 - 5)}
\]
\[
\text{Area} = \sqrt{10 \times 2 \times 3 \times 5}
\]
\[
\text{Area} = \sqrt{300}
\]
\[
\text{Area} \approx 17.3 \, \text{cm}^2 \quad (\text{correct to 1 decimal place})
\]
Answers for the triangle:
\[
\text{Semi-perimeter: } \boxed{10 \, \text{cm}}
\]
\[
\text{Area: } \boxed{17.3 \, \text{cm}^2}
\]
---
Final Answers:
1.
Square: \(\boxed{25 \, \text{cm}^2}\)
2.
Rectangle: \(\boxed{66 \, \text{m}^2}\)
3.
Right Triangle: \(\boxed{40 \, \text{cm}^2}\)
4.
Semi-perimeter of the triangle: \(\boxed{10 \, \text{cm}}\)
5.
Area of the triangle using Heron's formula: \(\boxed{17.3 \, \text{cm}^2}\)
Parent Tip: Review the logic above to help your child master the concept of area of rectangles and triangles worksheet.
