Find the perimeter and area of each figure below. a. b. 7 ft 5 ft 4 ft c. 15 cm 8 cm
The Correct Answer and Explanation is:
Figure b: A Triangle with sides 7 ft, 5 ft, and 4 ft
Perimeter:
The perimeter of a triangle is the sum of the lengths of all its sides. Perimeter=7+5+4=16 ft\text{Perimeter} = 7 + 5 + 4 = 16 \text{ ft}Perimeter=7+5+4=16 ft
Area:
To find the area of a triangle when all three sides are known, we use Heron’s formula:
- First, calculate the semi-perimeter:
s=7+5+42=162=8s = \frac{7 + 5 + 4}{2} = \frac{16}{2} = 8s=27+5+4=216=8
- Then use the formula:
A=s(s−a)(s−b)(s−c)A = \sqrt{s(s – a)(s – b)(s – c)}A=s(s−a)(s−b)(s−c) A=8(8−7)(8−5)(8−4)=8×1×3×4A = \sqrt{8(8 – 7)(8 – 5)(8 – 4)} = \sqrt{8 \times 1 \times 3 \times 4}A=8(8−7)(8−5)(8−4)=8×1×3×4 A=96≈9.80 ft2A = \sqrt{96} \approx 9.80 \text{ ft}^2A=96≈9.80 ft2
Figure c: A Rectangle with sides 15 cm and 8 cm
Perimeter:
Perimeter of a rectangle is: Perimeter=2×(15+8)=2×23=46 cm\text{Perimeter} = 2 \times (15 + 8) = 2 \times 23 = 46 \text{ cm}Perimeter=2×(15+8)=2×23=46 cm
Area:
Area of a rectangle is: Area=length×width=15×8=120 cm2\text{Area} = \text{length} \times \text{width} = 15 \times 8 = 120 \text{ cm}^2Area=length×width=15×8=120 cm2
Explanation:
Understanding how to calculate the perimeter and area of different geometric shapes is a foundational skill in geometry. Each shape has its own formula depending on its characteristics.
For figure b, we identified it as a triangle with sides of 7 feet, 5 feet, and 4 feet. To find its perimeter, we simply added the three sides together, which gave us 16 feet. However, finding the area required more thought. Since no height or angle was provided, we used Heron’s formula, which allows us to find the area of a triangle when all three sides are known. The process involves calculating the semi-perimeter first and then using it in a square root formula that multiplies the semi-perimeter by the difference between it and each side. The result was approximately 9.80 square feet.
In figure c, we saw a rectangle with dimensions 15 cm by 8 cm. Rectangles are simpler when it comes to area and perimeter. The perimeter is calculated by adding the length and width and then multiplying by 2 (since rectangles have two pairs of equal sides). The area is simply length times width, which makes it straightforward. In this case, the perimeter came out to 46 cm, and the area was 120 square centimeters.
These calculations are not only academically important but also practically useful in real-world applications like architecture, construction, and interior design. Understanding the formulas helps develop spatial awareness and problem-solving skills.
