Here are the step-by-step solutions for the problems on the worksheet.
Section A: Area of Triangles
The formula used for these questions is:
$$ \text{Area} = \frac{1}{2} \times a \times b \times \sin(C) $$
*(Where $a$ and $b$ are two sides, and $C$ is the angle between them)*
1)
*
Sides: 7 cm and 6 cm
*
Angle: $100^\circ$
*
Calculation: $0.5 \times 7 \times 6 \times \sin(100^\circ)$
*
Result: $20.69...$
*
Rounded (3 sig figs): 20.7 cm²
2)
*
Sides: 3.2 cm and 4.5 cm
*
Angle: $84^\circ$
*
Calculation: $0.5 \times 3.2 \times 4.5 \times \sin(84^\circ)$
*
Result: $7.160...$
*
Rounded (3 sig figs): 7.16 cm²
3)
*
Sides: 8.1 cm and 12.3 cm
*
Angle: $62^\circ$
*
Calculation: $0.5 \times 8.1 \times 12.3 \times \sin(62^\circ)$
*
Result: $44.11...$
*
Rounded (3 sig figs): 44.1 cm²
4)
*
Sides: 9 cm and 5 cm
*
Angle: $27^\circ$
*
Calculation: $0.5 \times 9 \times 5 \times \sin(27^\circ)$
*
Result: $10.21...$
*
Rounded (3 sig figs): 10.2 cm²
5)
*
Sides: 2.7 cm and 1.4 cm
*
Angle: $112^\circ$
*
Calculation: $0.5 \times 2.7 \times 1.4 \times \sin(112^\circ)$
*
Result: $1.749...$
*
Rounded (3 sig figs): 1.75 cm²
6) Find side $x$:
*
Given: Area = 31.7 cm², Side $a$ = 10 cm, Angle = $44^\circ$, Side $b$ = $x$
*
Formula rearranged: $x = \frac{\text{Area}}{0.5 \times 10 \times \sin(44^\circ)}$
*
Calculation: $31.7 \div (5 \times \sin(44^\circ))$
*
Result: $9.131...$
*
Rounded (3 sig figs): 9.13 cm
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Section B: Parallelograms and Other Shapes
1) Parallelogram
*
Formula: $\text{Area} = a \times b \times \sin(\theta)$ (Note: There is no $\frac{1}{2}$ for parallelograms).
*
Sides: 19.5 cm and 15 cm
*
Angle: $71^\circ$
*
Calculation: $19.5 \times 15 \times \sin(71^\circ)$
*
Result: $277.49...$
*
Rounded (3 sig figs): 277 cm²
2) Arrow-head shape
This shape is made of two triangles joined together. We calculate the area of each triangle separately and add them up.
*
Triangle 1 (Left): Sides 6 cm and 2 cm, Angle $128^\circ$.
* $\text{Area} = 0.5 \times 6 \times 2 \times \sin(128^\circ) = 4.728...$
*
Triangle 2 (Right): Sides 6 cm and 2 cm (radii of the circle), Angle $47^\circ$.
* $\text{Area} = 0.5 \times 6 \times 2 \times \sin(47^\circ) = 2.194...$
*
Total Area: $4.728 + 2.194 = 6.922...$
*
Rounded (3 sig figs): 6.92 cm²
3) Irregular Quadrilateral
We split this shape into two triangles using the dotted line.
*
Triangle 1 (Bottom Left - Right Angled): Base 3.6 cm, Height 1.9 cm.
* $\text{Area} = 0.5 \times 3.6 \times 1.9 = 3.42 \text{ cm}^2$
* We need the hypotenuse (dotted line) for the next part: $\sqrt{3.6^2 + 1.9^2} = 4.0706 \text{ cm}$.
*
Triangle 2 (Top): Sides 2.2 cm and 4.0706 cm (the dotted line), Angle $51^\circ$.
* $\text{Area} = 0.5 \times 2.2 \times 4.0706 \times \sin(51^\circ) = 3.482... \text{ cm}^2$
*
Total Area: $3.42 + 3.482... = 6.902...$
*
Rounded (3 sig figs): 6.90 cm²
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Final Answer:
Section A
1) 20.7 cm²
2) 7.16 cm²
3) 44.1 cm²
4) 10.2 cm²
5) 1.75 cm²
6) x = 9.13 cm
Section B
1) 277 cm²
2) 6.92 cm²
3) 6.90 cm²
Parent Tip: Review the logic above to help your child master the concept of triangle area worksheet.
