What is the area shaded in this figure?

Question

I was sent this problem by a friend, they say it is a year 6 question. Using trigonometry I got the shaded area as $4 - \frac{\pi}{3}$.

Now Year six, I doubt if they do trigonometry. Is there another approach out there?

Answer 1

Label the vertices as shown in the diagram. $\tan\alpha=\frac{1}{2}\implies \tan 2\alpha=\frac{4}{3}\implies 2\alpha=\tan^{-1}(\frac{4}{3})$

$\therefore$ Area of circular sector$ BAG=8\tan^{-1}(\frac{4}{3})$

Area of $\triangle AGC=16\cos\alpha\sin\alpha=8\sin2\alpha=\frac{32}{5}$

Thus, area of curved figure $BEG=\frac{48}{5}-8\tan^{-1}(\frac{4}{3})$

Therefore, Area of shaded part$=\frac{1}{2}$ (Area of $BEFC$)$-\frac{1}{2}$ (Area of semicircle)$-$Area of curved figure $BEG$.

This gives $\frac{32}{5}-4\pi+8\tan^{-1}(\frac{4}{3})=\frac{32}{5}-4\tan^{-1}(\frac{3}{4})\approx1.252$

Answer 2

Flip the image by vertically.

It is not as simple as you think. Year six is probably high school when they can do geometry and calculus. I have done it using calculus and the area is as shown in the figure

Answer 3

$\color{blue}{\text{Triangle Area }}$(the larger blue triangle) = $\color{blue}{\text{Sector Area }}$(of the circle) + one $\color{blue}{\text{Corner Area }}$+ $\color{red}{\text{x Area }}$

If radius (r) = 1

Steps	Formulas	Values
1.	$A_{\color{green}{\bigcirc}} = \pi r^2$	$\pi (1)^2 = \pi \times 1 = \pi \approx 3.1416$
2.	$A_{\color{brown}{\square}} = \text{Side}^2$	$2 \times 2 = 4$
3.	$A_{\color{blue}{\triangle}} = \frac{1}{2} \times \text{Base} \times \text{Height}$	$\frac{1}{2} \times 2 \times 1 = 1$
4.	$A_{\text{corner}} = \frac{A_{\color{brown}{\square}} - A_{\color{green}{\bigcirc}}}{4}$	$\frac{4 - 3.1415}{4} = 0.2146$

$A_{\text{corner}}$ is the area outside the circle but inside the square for a single corner only.

Step 5 – The BLUE Triangle | Find the $\alpha$

Using trigonometry, given that Opposite side (a) = 1 and Adjacent side (b) = 2

∴ $tan(\alpha)$ = 1/2 => $\alpha$ = $tan^{-1} \left( \frac{1}{2} \right)$ => $\alpha$ = 26.5650°

$\color{lightgray}{\boxed{\text{Hypotenuse } (c) \text{ is: } a^2 + b^2 = c^2 \Rightarrow 1^2 + 2^2 = c^2 \Rightarrow 1 + 4 = c^2 \Rightarrow \sqrt{5} = c \approx 2.236}}$

Step 6,7 – The ORANGE Triangle | Find the $\beta$ and Area $A_{\color{orange}{\triangle}}$

Intuitively analyze it, the Orange Triangle is Isosceles, and one of its angles is an alternate interior angle of $\alpha$.

∴ both of its Base Angles are 26.5650$^\circ$

∴ the Vertex Angle ($\beta$) = 180$^\circ$ - 26.565$^\circ$ - 26.565$^\circ$ $\approx$ 126.8700$^\circ$

Splitting the orange isosceles triangle from its center creates two equal right-angled triangles. By drawing a perpendicular from the vertex angle, or equivalently from the center of the circle to the base of the orange isosceles triangle, divide it into two congruent (or equal) right-angled triangles.

Now, consider The Orange Triangle as two right-angled triangles with the Hypotenuse (c) is equal to the radius (r) = 1 and $\alpha$ = 26.5650$^\circ$

Using Trigonometry

Opposite side (a) = c $\cdot sin$(26.5650$^\circ$) $\approx$ 0.4472
Adjacent side (b) = c $\cdot cos$(26.5650$^\circ$) $\approx$ 0.8944

Since, the area of The Orange Triangle is twice the sum of both halves

Area $A_{\color{orange}{\triangle}}$ = 2 $\times \frac{1}{2} \times b \times a$ $\Rightarrow$ 0.4472 $\times$ 0.8944 $\approx$ 0.4

Alternatively, a more common approach is to duplicate the orange isosceles triangle along its non radii side to form a parallelogram. The area of this parallelogram is given by
$A = r \times h$, where the height $h$ is $r \cdot \sin(2\alpha)$. The factor $2$ appears because the angle is also duplicated at this corner, angle $\beta$ doesn't have this issue. Taking half of this area gives the area of the orange triangle.

Step 8,9 – The Non-GREEN region of the Circle | Find Sector & Segment

Area of the minor sector of the circle using Vertex Angle ($\beta$)
$A_{\text{sector}} = \frac{A_{\color{green}{\bigcirc}}}{360} \times \beta$ $\Rightarrow$ $A_{\text{sector}} = \frac{3.1416}{360} \times$ 126.8700 $\approx$ 1.1072

By removing the orange triangle from this sector left with the area of the minor segment (or blue Segment)

$A_{\color{blue}{segment}}$ = $A_{\text{sector}}$ - $A_{\color{orange}{\triangle}}$ $\Rightarrow$ 1.1072 - 0.4 $\approx$ 0.7072

---

Step 10 – Find area X

$A_{\color{blue}{\triangle}}$ = $A_{\color{blue}{segment}}$ + $A_{\color{blue}{corner}}$ + $\color{red}{A}_{\color{red}{x}}$
$\Rightarrow$ $\color{red}{A}_{\color{red}{x}}$ = $A_{\color{blue}{\triangle}}$ - $A_{\color{blue}{segment}}$ - $A_{\color{blue}{corner}}$
$\Rightarrow$ $\color{red}{A}_{\color{red}{x}}$ = 1 - 0.7072 - 0.2146
$\Rightarrow$ $\color{red}{A}_{\color{red}{x}}$ = 0.0782

Note: ∵ r = 1, Ax = 0.0782. For r = 4 (diameter = 8), see the table below.

---

Summary

1.  - Area of the circle
2.  - Area of the circumscribed square
3.  - Area of the blue right-angle triangle (quarter of the square)
4.  - Area of the one Corner (quarter of the area of Square minus Circle)
5.  - Angle α of the blue triangle using tan(α) where a=radius and b=diameter
6.  - Angle β of the Orange Triangle, an isosceles triangle with two angles as α
7.  - Area of the Orange Triangle calculated using two right-angled triangles
8.  - Area of the minor sector using β
9.  - Area of the minor segment (sector minus Orange Triangle)
10. - AreaX = area of Blue triangle - area of segment - area of Corner

---

JavaScript

const diameters = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]; // Example diameters
const calculations = [];
function calculateProperties(diameter) {
    const radius = diameter / 2;
    const areaCircle = Math.PI * Math.pow(radius, 2);
    const areaSquare = Math.pow(diameter, 2);
    const areaBlueTriangle = areaSquare / 4;
    const areaCorner = (areaSquare - areaCircle) / 4;
const angleAlpha = Math.atan(radius / diameter) * (180 / Math.PI); // In degrees
const angleBeta = (180 - 2 * angleAlpha);

// Step 7: Calculate area of the Orange Isosceles Triangle using right-angled triangles
const x = radius * Math.cos(angleAlpha * Math.PI / 180);
const h = radius * Math.sin(angleAlpha * Math.PI / 180);
const areaHalfOrangeTriangle = 0.5 * x * h;
const areaOrangeTriangle = 2 * areaHalfOrangeTriangle;

// Step 8: Area of the minor sector (using Beta)
const areaMinorSector = (angleBeta / 360) * areaCircle;

// Step 9: Area of the minor segment (sector minus Orange Triangle)
const areaMinorSegment = areaMinorSector - areaOrangeTriangle;

// Step 10: AreaX = area of Blue triangle - area of segment - area of Corner
const areaX = areaBlueTriangle - areaMinorSegment - areaCorner;

return { 
    radius, 
    areaCircle, 
    areaSquare, 
    areaBlueTriangle, 
    angleAlpha, 
    angleBeta, 
    areaOrangeTriangle, 
    areaCorner, 
    areaMinorSector, 
    areaMinorSegment, 
    areaX
};

}
function printResults(results) {
    console.log("Calculated Values:");
    results.forEach((res, index) => {
        console.log(Input Diameter: ${diameters[index]});
        console.log(0. Radius: ${res.radius.toFixed(4)});
        console.log(1. Area of Circle: ${res.areaCircle.toFixed(4)});
        console.log(2. Area of Square: ${res.areaSquare.toFixed(4)});
        console.log(3. Area of Blue Triangle: ${res.areaBlueTriangle.toFixed(4)});
        console.log(4. Area of Corner: ${res.areaCorner.toFixed(4)});
        console.log(5. Angle α: ${res.angleAlpha.toFixed(4)}°);
        console.log(6. Angle β (Orange Triangle): ${res.angleBeta.toFixed(4)}°);
        console.log(7. Area of Orange Triangle: ${res.areaOrangeTriangle.toFixed(4)});
        console.log(8. Area of Minor Sector: ${res.areaMinorSector.toFixed(4)});
        console.log(9. Area of Minor Segment: ${res.areaMinorSegment.toFixed(4)});
        console.log(10. AreaX: ${res.areaX.toFixed(4)});
        console.log("------------------------------");
    });
}
diameters.forEach(d => calculations.push(calculateProperties(d)));
printResults(calculations);

Output:

Property	Diameter=1	Diameter=2	Diameter=3	Diameter=4	Diameter=5	Diameter=6	Diameter=7	Diameter=8	Diameter=9	Diameter=10	Diameter=11	Diameter=12	Diameter=13	Diameter=14	Diameter=15	Diameter=16	Diameter=17	Diameter=18	Diameter=19	Diameter=20
Radius	0.5000	1.0000	1.5000	2.0000	2.5000	3.0000	3.5000	4.0000	4.5000	5.0000	5.5000	6.0000	6.5000	7.0000	7.5000	8.0000	8.5000	9.0000	9.5000	10.0000
Area of Circle	0.7854	3.1416	7.0686	12.5664	19.6350	28.2743	38.4845	50.2655	63.6173	78.5398	95.0332	113.0973	132.7323	153.9380	176.7146	201.0619	226.9801	254.4690	283.5287	314.1593
Area of Square	1.0000	4.0000	9.0000	16.0000	25.0000	36.0000	49.0000	64.0000	81.0000	100.0000	121.0000	144.0000	169.0000	196.0000	225.0000	256.0000	289.0000	324.0000	361.0000	400.0000
Area of Blue Triangle	0.2500	1.0000	2.2500	4.0000	6.2500	9.0000	12.2500	16.0000	20.2500	25.0000	30.2500	36.0000	42.2500	49.0000	56.2500	64.0000	72.2500	81.0000	90.2500	100.0000
Area of Corner	0.0537	0.2146	0.4829	0.8584	1.3413	1.9314	2.6289	3.4336	4.3457	5.3650	6.4917	7.7257	9.0669	10.5155	12.0714	13.7345	15.5050	17.3827	19.3678	21.4602
Angle α	26.5651°	26.5651°	26.5651°	26.5651°	26.5651°	26.5651°	26.5651°	26.5651°	26.5651°	26.5651°	26.5651°	26.5651°	26.5651°	26.5651°	26.5651°	26.5651°	26.5651°	26.5651°	26.5651°	26.5651°
Angle β (Orange Triangle)	126.8699°	126.8699°	126.8699°	126.8699°	126.8699°	126.8699°	126.8699°	126.8699°	126.8699°	126.8699°	126.8699°	126.8699°	126.8699°	126.8699°	126.8699°	126.8699°	126.8699°	126.8699°	126.8699°	126.8699°
Area of Orange Triangle	0.1000	0.4000	0.9000	1.6000	2.5000	3.6000	4.9000	6.4000	8.1000	10.0000	12.1000	14.4000	16.9000	19.6000	22.5000	25.6000	28.9000	32.4000	36.1000	40.0000
Area of Minor Sector	0.2768	1.1071	2.4911	4.4286	6.9197	9.9643	13.5626	17.7144	22.4198	27.6787	33.4912	39.8574	46.7770	54.2503	62.2771	70.8575	79.9915	89.6790	99.9202	110.7149
Area of Minor Segment	0.1768	0.7071	1.5911	2.8286	4.4197	6.3643	8.6626	11.3144	14.3198	17.6787	21.3912	25.4574	29.8770	34.6503	39.7771	45.2575	51.0915	57.2790	63.8202	70.7149
AreaX	0.0196	0.0782	0.1761	0.3130	0.4891	0.7042	0.9586	1.2520 *	1.5846	1.9562	2.3670	2.8170	3.3060	3.8342	4.4015	5.0080	5.6535	6.3382	7.0620	7.8249