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Intermediate Geometry : How to find the area of an acute / obtuse isosceles triangle

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Example Question #1 : How To Find The Area Of An Acute / Obtuse Isosceles Triangle

An isosceles triangle has two legs of length 10" with a base of unknown length, and has a height of 6". Find the area.

Possible Answers:

$64\ in^2$ $\displaystyle 64\ in^2$

$32\ in^2$ $\displaystyle 32\ in^2$

$36\ in^2$ $\displaystyle 36\ in^2$

$60\ in^2$ $\displaystyle 60\ in^2$

$48\ in^2$ $\displaystyle 48\ in^2$

Correct answer:

$48\ in^2$ $\displaystyle 48\ in^2$

Explanation:

To find the area:

$\frac{1}{2}\cdot base\cdot height$ $\displaystyle \frac{1}{2}\cdot base\cdot height$

The information giving us the two sides helps us find the base.

Using the fact that an isosceles triangle can be split vertically down the middle (note: the length of this extra line will be equal to the height) to form two identical right triangles, we then use the Pythagorean Theorem to find the base:

$\displaystyle a^2 + b^2 = c^2$

For our problem, $\displaystyle a$ is the height, $\displaystyle b$ is the base (of ONE of the right triangles; the base of the isosceles triangle will be twice as big) and $\displaystyle c$ is the hypotenuse, or the leg of length 10".

We find that the base is 8", so the base of the isosceles triangle is 16".

Plugging in our numbers we get:

$16\cdot 6\cdot \frac{1}{2}=48\ in^{2}$ $\displaystyle 16\cdot 6\cdot \frac{1}{2}=48\ in^{2}$

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Example Question #2 : How To Find The Area Of An Acute / Obtuse Isosceles Triangle

An isosceles triangle is placed in a circle as shown by the figure below.

If diameter of the circle is $\displaystyle 2$, find the area of the shaded region.

Possible Answers:

4.22 $\displaystyle 4.22$

2.14 $\displaystyle 2.14$

1.93 $\displaystyle 1.93$

3.19 $\displaystyle 3.19$

Correct answer:

2.14 $\displaystyle 2.14$

Explanation:

From the given image, you should notice that the base of the triangle is also the diameter of the circle. In addition, the height of the triangle is also the radius of the circle.

Thus, we can find the area of the triangle.

$\text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})$ $\displaystyle \text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})$

$\text{Area of Triangle}=\frac{1}{2}(2 \times 1)=1$ $\displaystyle \text{Area of Triangle}=\frac{1}{2}(2 \times 1)=1$

Next, recall how to find the area of a circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$ $\displaystyle \text{Area of Circle}=\pi\times\text{radius}^2$

$\text{Area of Circle}=\pi\times 1^2= \pi$ $\displaystyle \text{Area of Circle}=\pi\times 1^2= \pi$

To find the area of the shaded region, subtract the two areas.

$\text{Area of Shaded Region}=\text{Area of Circle}-\text{Area of Triangle}$ $\displaystyle \text{Area of Shaded Region}=\text{Area of Circle}-\text{Area of Triangle}$

$\text{Area of Shaded Region}=\pi-1=2.14$ $\displaystyle \text{Area of Shaded Region}=\pi-1=2.14$

Make sure to round to $\displaystyle 2$ places after the decimal.

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Example Question #3 : How To Find The Area Of An Acute / Obtuse Isosceles Triangle

An isosceles triangle is placed in a circle as shown by the figure below.

If the diameter of the circle is $\displaystyle 12$, find the area of the shaded region.

Possible Answers:

75.59 $\displaystyle 75.59$

91.21 $\displaystyle 91.21$

77.10 $\displaystyle 77.10$

60.92 $\displaystyle 60.92$

Correct answer:

77.10 $\displaystyle 77.10$

Explanation:

From the given image, you should notice that the base of the triangle is also the diameter of the circle. In addition, the height of the triangle is also the radius of the circle.

Thus, we can find the area of the triangle.

$\text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})$ $\displaystyle \text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})$

$\text{Area of Triangle}=\frac{1}{2}(12 \times 6)=36$ $\displaystyle \text{Area of Triangle}=\frac{1}{2}(12 \times 6)=36$

Next, recall how to find the area of a circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$ $\displaystyle \text{Area of Circle}=\pi\times\text{radius}^2$

$\text{Area of Circle}=\pi\times 6^2= 36\pi$ $\displaystyle \text{Area of Circle}=\pi\times 6^2= 36\pi$

To find the area of the shaded region, subtract the two areas.

$\text{Area of Shaded Region}=\text{Area of Circle}-\text{Area of Triangle}$ $\displaystyle \text{Area of Shaded Region}=\text{Area of Circle}-\text{Area of Triangle}$

$\text{Area of Shaded Region}=36\pi-36=77.10$ $\displaystyle \text{Area of Shaded Region}=36\pi-36=77.10$

Make sure to round to $\displaystyle 2$ places after the decimal.

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Example Question #4 : How To Find The Area Of An Acute / Obtuse Isosceles Triangle

An isosceles triangle is placed in a circle as shown by the figure below.

If the diameter of the circle is $\displaystyle 14$, find the area of the shaded region.

Possible Answers:

98.87 $\displaystyle 98.87$

102.05 $\displaystyle 102.05$

104.94 $\displaystyle 104.94$

112.52 $\displaystyle 112.52$

Correct answer:

104.94 $\displaystyle 104.94$

Explanation:

From the given image, you should notice that the base of the triangle is also the diameter of the circle. In addition, the height of the triangle is also the radius of the circle.

Thus, we can find the area of the triangle.

$\text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})$ $\displaystyle \text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})$

$\text{Area of Triangle}=\frac{1}{2}(14 \times 7)=49$ $\displaystyle \text{Area of Triangle}=\frac{1}{2}(14 \times 7)=49$

Next, recall how to find the area of a circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$ $\displaystyle \text{Area of Circle}=\pi\times\text{radius}^2$

$\text{Area of Circle}=\pi\times 7^2=49 \pi$ $\displaystyle \text{Area of Circle}=\pi\times 7^2=49 \pi$

To find the area of the shaded region, subtract the two areas.

$\text{Area of Shaded Region}=\text{Area of Circle}-\text{Area of Triangle}$ $\displaystyle \text{Area of Shaded Region}=\text{Area of Circle}-\text{Area of Triangle}$

$\text{Area of Shaded Region}=49\pi-49=104.94$ $\displaystyle \text{Area of Shaded Region}=49\pi-49=104.94$

Make sure to round to $\displaystyle 2$ places after the decimal.

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Example Question #5 : How To Find The Area Of An Acute / Obtuse Isosceles Triangle

An isosceles triangle is placed in a circle as shown by the figure below.

If the diameter of the circle is $\displaystyle 8$, find the area of the shaded region.

Possible Answers:

34.27 $\displaystyle 34.27$

40.29 $\displaystyle 40.29$

31.02 $\displaystyle 31.02$

29.95 $\displaystyle 29.95$

Correct answer:

34.27 $\displaystyle 34.27$

Explanation:

From the given image, you should notice that the base of the triangle is also the diameter of the circle. In addition, the height of the triangle is also the radius of the circle.

Thus, we can find the area of the triangle.

$\text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})$ $\displaystyle \text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})$

$\text{Area of Triangle}=\frac{1}{2}(8 \times 4)=16$ $\displaystyle \text{Area of Triangle}=\frac{1}{2}(8 \times 4)=16$

Next, recall how to find the area of a circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$ $\displaystyle \text{Area of Circle}=\pi\times\text{radius}^2$

$\text{Area of Circle}=\pi\times 4^2= 16\pi$ $\displaystyle \text{Area of Circle}=\pi\times 4^2= 16\pi$

To find the area of the shaded region, subtract the two areas.

$\text{Area of Shaded Region}=\text{Area of Circle}-\text{Area of Triangle}$ $\displaystyle \text{Area of Shaded Region}=\text{Area of Circle}-\text{Area of Triangle}$

$\text{Area of Shaded Region}=16\pi-16=34.27$ $\displaystyle \text{Area of Shaded Region}=16\pi-16=34.27$

Make sure to round to $\displaystyle 2$ places after the decimal.

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Example Question #6 : How To Find The Area Of An Acute / Obtuse Isosceles Triangle

An isosceles triangle is placed in a circle as shown by the figure below.

If the diameter of the circle is $\displaystyle 20$, find the area of the shaded region.

Possible Answers:

222.49 $\displaystyle 222.49$

328.52 $\displaystyle 328.52$

214.16 $\displaystyle 214.16$

190.06 $\displaystyle 190.06$

Correct answer:

214.16 $\displaystyle 214.16$

Explanation:

From the given image, you should notice that the base of the triangle is also the diameter of the circle. In addition, the height of the triangle is also the radius of the circle.

Thus, we can find the area of the triangle.

$\text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})$ $\displaystyle \text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})$

$\text{Area of Triangle}=\frac{1}{2}(20 \times 10)=100$ $\displaystyle \text{Area of Triangle}=\frac{1}{2}(20 \times 10)=100$

Next, recall how to find the area of a circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$ $\displaystyle \text{Area of Circle}=\pi\times\text{radius}^2$

$\text{Area of Circle}=\pi\times 10^2=100 \pi$ $\displaystyle \text{Area of Circle}=\pi\times 10^2=100 \pi$

To find the area of the shaded region, subtract the two areas.

$\text{Area of Shaded Region}=\text{Area of Circle}-\text{Area of Triangle}$ $\displaystyle \text{Area of Shaded Region}=\text{Area of Circle}-\text{Area of Triangle}$

$\text{Area of Shaded Region}=100\pi-100=214.16$ $\displaystyle \text{Area of Shaded Region}=100\pi-100=214.16$

Make sure to round to $\displaystyle 2$ places after the decimal.

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Example Question #7 : How To Find The Area Of An Acute / Obtuse Isosceles Triangle

An isosceles triangle is placed in a circle as shown by the figure below.

If the diameter of the circle is $\displaystyle 28$, find the area of the shaded region.

Possible Answers:

420.94 $\displaystyle 420.94$

419.75 $\displaystyle 419.75$

398.50 $\displaystyle 398.50$

458.61 $\displaystyle 458.61$

Correct answer:

419.75 $\displaystyle 419.75$

Explanation:

From the given image, you should notice that the base of the triangle is also the diameter of the circle. In addition, the height of the triangle is also the radius of the circle.

Thus, we can find the area of the triangle.

$\text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})$ $\displaystyle \text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})$

$\text{Area of Triangle}=\frac{1}{2}(28 \times 14)=196$ $\displaystyle \text{Area of Triangle}=\frac{1}{2}(28 \times 14)=196$

Next, recall how to find the area of a circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$ $\displaystyle \text{Area of Circle}=\pi\times\text{radius}^2$

$\text{Area of Circle}=\pi\times 14^2=196 \pi$ $\displaystyle \text{Area of Circle}=\pi\times 14^2=196 \pi$

To find the area of the shaded region, subtract the two areas.

$\text{Area of Shaded Region}=\text{Area of Circle}-\text{Area of Triangle}$ $\displaystyle \text{Area of Shaded Region}=\text{Area of Circle}-\text{Area of Triangle}$

$\text{Area of Shaded Region}=196\pi-196=419.75$ $\displaystyle \text{Area of Shaded Region}=196\pi-196=419.75$

Make sure to round to $\displaystyle 2$ places after the decimal.

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Example Question #8 : How To Find The Area Of An Acute / Obtuse Isosceles Triangle

An isosceles triangle is placed in a circle as shown by the figure below.

If the diameter of the circle is $\displaystyle 18$, find the area of the shaded region.

Possible Answers:

173.47 $\displaystyle 173.47$

198.52 $\displaystyle 198.52$

174.55 $\displaystyle 174.55$

189.62 $\displaystyle 189.62$

Correct answer:

173.47 $\displaystyle 173.47$

Explanation:

From the given image, you should notice that the base of the triangle is also the diameter of the circle. In addition, the height of the triangle is also the radius of the circle.

Thus, we can find the area of the triangle.

$\text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})$ $\displaystyle \text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})$

$\text{Area of Triangle}=\frac{1}{2}(18 \times 9)=81$ $\displaystyle \text{Area of Triangle}=\frac{1}{2}(18 \times 9)=81$

Next, recall how to find the area of a circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$ $\displaystyle \text{Area of Circle}=\pi\times\text{radius}^2$

$\text{Area of Circle}=\pi\times 9^2= 81\pi$ $\displaystyle \text{Area of Circle}=\pi\times 9^2= 81\pi$

To find the area of the shaded region, subtract the two areas.

$\text{Area of Shaded Region}=\text{Area of Circle}-\text{Area of Triangle}$ $\displaystyle \text{Area of Shaded Region}=\text{Area of Circle}-\text{Area of Triangle}$

$\text{Area of Shaded Region}=81\pi-81=173.47$ $\displaystyle \text{Area of Shaded Region}=81\pi-81=173.47$

Make sure to round to $\displaystyle 2$ places after the decimal.

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Example Question #9 : How To Find The Area Of An Acute / Obtuse Isosceles Triangle

An isosceles triangle is placed in a circle as shown by the figure below.

If the diameter of the circle is $\displaystyle 26$, find the area of the shaded region.

Possible Answers:

401.25 $\displaystyle 401.25$

355.21 $\displaystyle 355.21$

372.90 $\displaystyle 372.90$

361.93 $\displaystyle 361.93$

Correct answer:

361.93 $\displaystyle 361.93$

Explanation:

From the given image, you should notice that the base of the triangle is also the diameter of the circle. In addition, the height of the triangle is also the radius of the circle.

Thus, we can find the area of the triangle.

$\text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})$ $\displaystyle \text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})$

$\text{Area of Triangle}=\frac{1}{2}(26 \times 13)=169$ $\displaystyle \text{Area of Triangle}=\frac{1}{2}(26 \times 13)=169$

Next, recall how to find the area of a circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$ $\displaystyle \text{Area of Circle}=\pi\times\text{radius}^2$

$\text{Area of Circle}=\pi\times 13^2= 169\pi$ $\displaystyle \text{Area of Circle}=\pi\times 13^2= 169\pi$

To find the area of the shaded region, subtract the two areas.

$\text{Area of Shaded Region}=\text{Area of Circle}-\text{Area of Triangle}$ $\displaystyle \text{Area of Shaded Region}=\text{Area of Circle}-\text{Area of Triangle}$

$\text{Area of Shaded Region}=169\pi-169=361.93$ $\displaystyle \text{Area of Shaded Region}=169\pi-169=361.93$

Make sure to round to $\displaystyle 2$ places after the decimal.

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Example Question #10 : How To Find The Area Of An Acute / Obtuse Isosceles Triangle

An isosceles triangle is placed in a circle as shown by the figure below.

If the diameter of the circle is $\displaystyle 44$, find the area of the shaded region.

Possible Answers:

1121.77 $\displaystyle 1121.77$

980.27 $\displaystyle 980.27$

1092.14 $\displaystyle 1092.14$

1036.53 $\displaystyle 1036.53$

Correct answer:

1036.53 $\displaystyle 1036.53$

Explanation:

From the given image, you should notice that the base of the triangle is also the diameter of the circle. In addition, the height of the triangle is also the radius of the circle.

Thus, we can find the area of the triangle.

$\text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})$ $\displaystyle \text{Area of Triangle}=\frac{1}{2}(\text{base}\times\text{height})$

$\text{Area of Triangle}=\frac{1}{2}(44 \times 22)=484$ $\displaystyle \text{Area of Triangle}=\frac{1}{2}(44 \times 22)=484$

Next, recall how to find the area of a circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$ $\displaystyle \text{Area of Circle}=\pi\times\text{radius}^2$

$\text{Area of Circle}=\pi\times 22^2= 484\pi$ $\displaystyle \text{Area of Circle}=\pi\times 22^2= 484\pi$

To find the area of the shaded region, subtract the two areas.

$\text{Area of Shaded Region}=\text{Area of Circle}-\text{Area of Triangle}$ $\displaystyle \text{Area of Shaded Region}=\text{Area of Circle}-\text{Area of Triangle}$

$\text{Area of Shaded Region}=484\pi-484=1036.53$ $\displaystyle \text{Area of Shaded Region}=484\pi-484=1036.53$

Make sure to round to $\displaystyle 2$ places after the decimal.

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Intermediate Geometry : How to find the area of an acute / obtuse isosceles triangle

Study concepts, example questions & explanations for Intermediate Geometry

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Example Questions

Example Question #1 : How To Find The Area Of An Acute / Obtuse Isosceles Triangle

Example Question #2 : How To Find The Area Of An Acute / Obtuse Isosceles Triangle

Example Question #3 : How To Find The Area Of An Acute / Obtuse Isosceles Triangle

Example Question #4 : How To Find The Area Of An Acute / Obtuse Isosceles Triangle

Example Question #5 : How To Find The Area Of An Acute / Obtuse Isosceles Triangle

Example Question #6 : How To Find The Area Of An Acute / Obtuse Isosceles Triangle

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Example Question #8 : How To Find The Area Of An Acute / Obtuse Isosceles Triangle

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Example Question #10 : How To Find The Area Of An Acute / Obtuse Isosceles Triangle

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