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Basic Geometry : How to find the area of a 45/45/90 right isosceles triangle

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Example Question #1085 : Plane Geometry

Find the area of the triangle if the radius of the circle is

Possible Answers:

Correct answer:

Explanation:

The two tick marks on the image indicate that those sides are congruent; therefore, this is an isosceles right triangle.

Notice that the hypotenuse of the triangle is also the diameter of the circle. Recall the relationship between the diameter of a circle and its radius:

$\text{Hypotenuse}=\text{Diameter}=2\times\text{radius}$

Substitute in the value of the radius to find the length of the diameter.

$\text{Hypotenuse}=2\times 4$

Simplify.

$\text{Hypotenuse}=8$

Now, use the Pythagorean theorem to find the lengths of the missing sides of the triangle.

$\text{side}^2+\text{side}^2=\text{hypotenuse}^2$

$2(\text{side})^2=\text{Hypotenuse}^2$

$\text{side}^2=\frac{\text{Hypotenuse}^2}{2}$

Now, recall how to find the area of a triangle:

$\text{Area}=\frac{\text{base}\times\text{height}}{2}$

In this case, because we have an isosceles right triangle,

$\text{base}=\text{height}=\text{side}$

$\text{Area}=\frac{\text{side}^2}{2}$

Take the equation derived from the Pythagorean theorem and plug it in to the equation above.

$\text{Area}=\frac{\text{Hypotenuse}^2}{2}\times \frac{1}{2}$

Simplify.

$\text{Area}=\frac{\text{Hypotenuse}^2}{4}$

Now, substitute in the value of the hypotenuse to find the area of the triangle.

$\text{Area}=\frac{8^2}{4}$

Solve.

$\text{Area}=16$

Report an Error

Example Question #1086 : Plane Geometry

Find the area of the triangle if the radius of the circle is .

Possible Answers:

$\frac{1}{2}$

Correct answer:

Explanation:

The two tick marks on the image indicate that those sides are congruent; therefore, this is an isosceles right triangle.

Notice that the hypotenuse of the triangle is also the diameter of the circle. Recall the relationship between the diameter of a circle and its radius:

$\text{Hypotenuse}=\text{Diameter}=2\times\text{radius}$

Substitute in the value of the radius to find the length of the diameter.

$\text{Hypotenuse}=2\times 2$

Simplify.

$\text{Hypotenuse}=4$

Now, use the Pythagorean theorem to find the lengths of the missing sides of the triangle.

$\text{side}^2+\text{side}^2=\text{hypotenuse}^2$

$2(\text{side})^2=\text{Hypotenuse}^2$

$\text{side}^2=\frac{\text{Hypotenuse}^2}{2}$

Now, recall how to find the area of a triangle:

$\text{Area}=\frac{\text{base}\times\text{height}}{2}$

In this case, because we have an isosceles right triangle,

$\text{base}=\text{height}=\text{side}$

$\text{Area}=\frac{\text{side}^2}{2}$

Take the equation derived from the Pythagorean theorem and plug it in to the equation above.

$\text{Area}=\frac{\text{Hypotenuse}^2}{2}\times \frac{1}{2}$

Simplify.

$\text{Area}=\frac{\text{Hypotenuse}^2}{4}$

Now, substitute in the value of the hypotenuse to find the area of the triangle.

$\text{Area}=\frac{4^2}{4}$

Solve.

$\text{Area}=4$

Report an Error

Example Question #101 : Triangles

Find the area of the triangle if the radius of the circle is .

Possible Answers:

$\frac{1}{2}$

Correct answer:

Explanation:

The two tick marks on the image indicate that those sides are congruent; therefore, this is an isosceles right triangle.

Notice that the hypotenuse of the triangle is also the diameter of the circle. Recall the relationship between the diameter of a circle and its radius:

$\text{Hypotenuse}=\text{Diameter}=2\times\text{radius}$

Substitute in the value of the radius to find the length of the diameter.

$\text{Hypotenuse}=2\times 1$

Simplify.

$\text{Hypotenuse}=2$

Now, use the Pythagorean theorem to find the lengths of the missing sides of the triangle.

$\text{side}^2+\text{side}^2=\text{hypotenuse}^2$

$2(\text{side})^2=\text{Hypotenuse}^2$

$\text{side}^2=\frac{\text{Hypotenuse}^2}{2}$

Now, recall how to find the area of a triangle:

$\text{Area}=\frac{\text{base}\times\text{height}}{2}$

In this case, because we have an isosceles right triangle,

$\text{base}=\text{height}=\text{side}$

$\text{Area}=\frac{\text{side}^2}{2}$

Take the equation derived from the Pythagorean theorem and plug it in to the equation above.

$\text{Area}=\frac{\text{Hypotenuse}^2}{2}\times \frac{1}{2}$

Simplify.

$\text{Area}=\frac{\text{Hypotenuse}^2}{4}$

Now, substitute in the value of the hypotenuse to find the area of the triangle.

$\text{Area}=\frac{2^2}{4}$

Solve.

$\text{Area}=1$

Report an Error

Example Question #31 : How To Find The Area Of A 45/45/90 Right Isosceles Triangle

Find the area of the triangle if the radius of the circle is .

Possible Answers:

125

100

Correct answer:

Explanation:

The two tick marks on the image indicate that those sides are congruent; therefore, this is an isosceles right triangle.

Notice that the hypotenuse of the triangle is also the diameter of the circle. Recall the relationship between the diameter of a circle and its radius:

$\text{Hypotenuse}=\text{Diameter}=2\times\text{radius}$

Substitute in the value of the radius to find the length of the diameter.

$\text{Hypotenuse}=2\times 5$

Simplify.

$\text{Hypotenuse}=10$

Now, use the Pythagorean theorem to find the lengths of the missing sides of the triangle.

$\text{side}^2+\text{side}^2=\text{hypotenuse}^2$

$2(\text{side})^2=\text{Hypotenuse}^2$

$\text{side}^2=\frac{\text{Hypotenuse}^2}{2}$

Now, recall how to find the area of a triangle:

$\text{Area}=\frac{\text{base}\times\text{height}}{2}$

In this case, because we have an isosceles right triangle,

$\text{base}=\text{height}=\text{side}$

$\text{Area}=\frac{\text{side}^2}{2}$

Take the equation derived from the Pythagorean theorem and plug it in to the equation above.

$\text{Area}=\frac{\text{Hypotenuse}^2}{2}\times \frac{1}{2}$

Simplify.

$\text{Area}=\frac{\text{Hypotenuse}^2}{4}$

Now, substitute in the value of the hypotenuse to find the area of the triangle.

$\text{Area}=\frac{10^2}{4}$

Solve.

$\text{Area}=25$

Report an Error

Example Question #101 : 45/45/90 Right Isosceles Triangles

Find the area of the triangle if the radius of the circle is .

Possible Answers:

144

Correct answer:

Explanation:

The two tick marks on the image indicate that those sides are congruent; therefore, this is an isosceles right triangle.

Notice that the hypotenuse of the triangle is also the diameter of the circle. Recall the relationship between the diameter of a circle and its radius:

$\text{Hypotenuse}=\text{Diameter}=2\times\text{radius}$

Substitute in the value of the radius to find the length of the diameter.

$\text{Hypotenuse}=2 \times6$

Simplify.

$\text{Hypotenuse}=12$

Now, use the Pythagorean theorem to find the lengths of the missing sides of the triangle.

$\text{side}^2+\text{side}^2=\text{hypotenuse}^2$

$2(\text{side})^2=\text{Hypotenuse}^2$

$\text{side}^2=\frac{\text{Hypotenuse}^2}{2}$

Now, recall how to find the area of a triangle:

$\text{Area}=\frac{\text{base}\times\text{height}}{2}$

In this case, because we have an isosceles right triangle,

$\text{base}=\text{height}=\text{side}$

$\text{Area}=\frac{\text{side}^2}{2}$

Take the equation derived from the Pythagorean theorem and plug it in to the equation above.

$\text{Area}=\frac{\text{Hypotenuse}^2}{2}\times \frac{1}{2}$

Simplify.

$\text{Area}=\frac{\text{Hypotenuse}^2}{4}$

Now, substitute in the value of the hypotenuse to find the area of the triangle.

$\text{Area}=\frac{12^2}{4}$

Solve.

$\text{Area}=36$

Report an Error

Example Question #102 : 45/45/90 Right Isosceles Triangles

Find the area of the triangle if the radius of the circle is .

Possible Answers:

Correct answer:

Explanation:

The two tick marks on the image indicate that those sides are congruent; therefore, this is an isosceles right triangle.

Notice that the hypotenuse of the triangle is also the diameter of the circle. Recall the relationship between the diameter of a circle and its radius:

$\text{Hypotenuse}=\text{Diameter}=2\times\text{radius}$

Substitute in the value of the radius to find the length of the diameter.

$\text{Hypotenuse}=2\times 7$

Simplify.

$\text{Hypotenuse}=14$

Now, use the Pythagorean theorem to find the lengths of the missing sides of the triangle.

$\text{side}^2+\text{side}^2=\text{hypotenuse}^2$

$2(\text{side})^2=\text{Hypotenuse}^2$

$\text{side}^2=\frac{\text{Hypotenuse}^2}{2}$

Now, recall how to find the area of a triangle:

$\text{Area}=\frac{\text{base}\times\text{height}}{2}$

In this case, because we have an isosceles right triangle,

$\text{base}=\text{height}=\text{side}$

$\text{Area}=\frac{\text{side}^2}{2}$

Take the equation derived from the Pythagorean theorem and plug it in to the equation above.

$\text{Area}=\frac{\text{Hypotenuse}^2}{2}\times \frac{1}{2}$

Simplify.

$\text{Area}=\frac{\text{Hypotenuse}^2}{4}$

Now, substitute in the value of the hypotenuse to find the area of the triangle.

$\text{Area}=\frac{14^2}{4}$

Solve.

$\text{Area}=49$

Report an Error

Example Question #111 : Triangles

Find the area of the triangle if the radius of the circle is .

Possible Answers:

$64\pi$

128

256

Correct answer:

Explanation:

The two tick marks on the image indicate that those sides are congruent; therefore, this is an isosceles right triangle.

Notice that the hypotenuse of the triangle is also the diameter of the circle. Recall the relationship between the diameter of a circle and its radius:

$\text{Hypotenuse}=\text{Diameter}=2\times\text{radius}$

Substitute in the value of the radius to find the length of the diameter.

$\text{Hypotenuse}=2\times 8$

Simplify.

$\text{Hypotenuse}=16$

Now, use the Pythagorean theorem to find the lengths of the missing sides of the triangle.

$\text{side}^2+\text{side}^2=\text{hypotenuse}^2$

$2(\text{side})^2=\text{Hypotenuse}^2$

$\text{side}^2=\frac{\text{Hypotenuse}^2}{2}$

Now, recall how to find the area of a triangle:

$\text{Area}=\frac{\text{base}\times\text{height}}{2}$

In this case, because we have an isosceles right triangle,

$\text{base}=\text{height}=\text{side}$

$\text{Area}=\frac{\text{side}^2}{2}$

Take the equation derived from the Pythagorean theorem and plug it in to the equation above.

$\text{Area}=\frac{\text{Hypotenuse}^2}{2}\times \frac{1}{2}$

Simplify.

$\text{Area}=\frac{\text{Hypotenuse}^2}{4}$

Now, substitute in the value of the hypotenuse to find the area of the triangle.

$\text{Area}=\frac{16^2}{4}$

Solve.

$\text{Area}=64$

Report an Error

Example Question #112 : 45/45/90 Right Isosceles Triangles

Find the area of the triangle if the radius of the circle is $\frac{1}{2}$ .

Possible Answers:

$\frac{1}{2}$

$\frac{1}{4}$

$\frac{1}{8}$

$\frac{1}{16}$

Correct answer:

$\frac{1}{4}$

Explanation:

The two tick marks on the image indicate that those sides are congruent; therefore, this is an isosceles right triangle.

Notice that the hypotenuse of the triangle is also the diameter of the circle. Recall the relationship between the diameter of a circle and its radius:

$\text{Hypotenuse}=\text{Diameter}=2\times\text{radius}$

Substitute in the value of the radius to find the length of the diameter.

$\text{Hypotenuse}=2\times \frac{1}{2}$

Simplify.

$\text{Hypotenuse}=1$

Now, use the Pythagorean theorem to find the lengths of the missing sides of the triangle.

$\text{side}^2+\text{side}^2=\text{hypotenuse}^2$

$2(\text{side})^2=\text{Hypotenuse}^2$

$\text{side}^2=\frac{\text{Hypotenuse}^2}{2}$

Now, recall how to find the area of a triangle:

$\text{Area}=\frac{\text{base}\times\text{height}}{2}$

In this case, because we have an isosceles right triangle,

$\text{base}=\text{height}=\text{side}$

$\text{Area}=\frac{\text{side}^2}{2}$

Take the equation derived from the Pythagorean theorem and plug it in to the equation above.

$\text{Area}=\frac{\text{Hypotenuse}^2}{2}\times \frac{1}{2}$

Simplify.

$\text{Area}=\frac{\text{Hypotenuse}^2}{4}$

Now, substitute in the value of the hypotenuse to find the area of the triangle.

$\text{Area}=\frac{1^2}{4}$

Solve.

$\text{Area}=\frac{1}{4}$

Report an Error

Example Question #113 : 45/45/90 Right Isosceles Triangles

Find the area of the triangle if the radius of the circle is $\frac{3}{2}$ .

Possible Answers:

$\frac{9}{8}$

$\frac{9}{2}$

$\frac{9}{4}$

Correct answer:

$\frac{9}{4}$

Explanation:

The two tick marks on the image indicate that those sides are congruent; therefore, this is an isosceles right triangle.

Notice that the hypotenuse of the triangle is also the diameter of the circle. Recall the relationship between the diameter of a circle and its radius:

$\text{Hypotenuse}=\text{Diameter}=2\times\text{radius}$

Substitute in the value of the radius to find the length of the diameter.

$\text{Hypotenuse}=2\times \frac{3}{2}$

Simplify.

$\text{Hypotenuse}=3$

Now, use the Pythagorean theorem to find the lengths of the missing sides of the triangle.

$\text{side}^2+\text{side}^2=\text{hypotenuse}^2$

$2(\text{side})^2=\text{Hypotenuse}^2$

$\text{side}^2=\frac{\text{Hypotenuse}^2}{2}$

Now, recall how to find the area of a triangle:

$\text{Area}=\frac{\text{base}\times\text{height}}{2}$

In this case, because we have an isosceles right triangle,

$\text{base}=\text{height}=\text{side}$

$\text{Area}=\frac{\text{side}^2}{2}$

Take the equation derived from the Pythagorean theorem and plug it in to the equation above.

$\text{Area}=\frac{\text{Hypotenuse}^2}{2}\times \frac{1}{2}$

Simplify.

$\text{Area}=\frac{\text{Hypotenuse}^2}{4}$

Now, substitute in the value of the hypotenuse to find the area of the triangle.

$\text{Area}=\frac{3^2}{4}$

Solve.

$\text{Area}=\frac{9}{4}$

Report an Error

Example Question #114 : 45/45/90 Right Isosceles Triangles

Find the area of the triangle if the radius of the circle is $\frac{9}{2}$ .

Possible Answers:

$\frac{81}{4}$

$\frac{81}{2}$

Correct answer:

$\frac{81}{4}$

Explanation:

The two tick marks on the image indicate that those sides are congruent; therefore, this is an isosceles right triangle.

Notice that the hypotenuse of the triangle is also the diameter of the circle. Recall the relationship between the diameter of a circle and its radius:

$\text{Hypotenuse}=\text{Diameter}=2\times\text{radius}$

Substitute in the value of the radius to find the length of the diameter.

$\text{Hypotenuse}=2\times \frac{9}{2}$

Simplify.

$\text{Hypotenuse}=9$

Now, use the Pythagorean theorem to find the lengths of the missing sides of the triangle.

$\text{side}^2+\text{side}^2=\text{hypotenuse}^2$

$2(\text{side})^2=\text{Hypotenuse}^2$

$\text{side}^2=\frac{\text{Hypotenuse}^2}{2}$

Now, recall how to find the area of a triangle:

$\text{Area}=\frac{\text{base}\times\text{height}}{2}$

In this case, because we have an isosceles right triangle,

$\text{base}=\text{height}=\text{side}$

$\text{Area}=\frac{\text{side}^2}{2}$

Take the equation derived from the Pythagorean theorem and plug it in to the equation above.

$\text{Area}=\frac{\text{Hypotenuse}^2}{2}\times \frac{1}{2}$

Simplify.

$\text{Area}=\frac{\text{Hypotenuse}^2}{4}$

Now, substitute in the value of the hypotenuse to find the area of the triangle.

$\text{Area}=\frac{9^2}{4}$

Solve.

$\text{Area}=\frac{81}{4}$

Report an Error

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Basic Geometry : How to find the area of a 45/45/90 right isosceles triangle

Study concepts, example questions & explanations for Basic Geometry

All Basic Geometry Resources

Example Questions

Example Question #1085 : Plane Geometry

Example Question #1086 : Plane Geometry

Example Question #101 : Triangles

Example Question #31 : How To Find The Area Of A 45/45/90 Right Isosceles Triangle

Example Question #101 : 45/45/90 Right Isosceles Triangles

Example Question #102 : 45/45/90 Right Isosceles Triangles

Example Question #111 : Triangles

Example Question #112 : 45/45/90 Right Isosceles Triangles

Example Question #113 : 45/45/90 Right Isosceles Triangles

Example Question #114 : 45/45/90 Right Isosceles Triangles

All Basic Geometry Resources

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