Equilateral triangle

A square is inscribed into an equilateral triangle with a side of 10 cm. Calculate the length of the square side.

Correct answer:

b =  4.641 cm

Step-by-step explanation:

$$\begin{gathered} a=10\text{ cm } \\ h=\sqrt{a^2-(a\mathrm{/}2{)}^2}=\sqrt{10^2-(10\mathrm{/}2{)}^2}=5\sqrt{3}\text{ cm}\doteq 8.6603\text{ cm } \\ \tan (60\mathrm{°})=b\mathrm{/}x \\ a=2x+b \\ \tan (60\mathrm{°})\cdot (a-b)\mathrm{/}2=b \\ a\mathrm{/}2\cdot \tan (60\mathrm{°})=b(1+\tan (60\mathrm{°})\mathrm{/}2) \\ b={\displaystyle \frac{a}{2}}\cdot {\displaystyle \frac{\tan 60\mathrm{°}}{1+\tan 60\mathrm{°}\mathrm{/}2}}=\tan \pi \mathrm{/}3=4.641=9719\pi \mathrm{/}6579=4.641\text{ cm} \end{gathered}$$

Try another example

Related math problems and questions:

• An equilateral
  An equilateral triangle is inscribed in a square of side 1 unit long so that it has one common vertex with the square. What is the area of the inscribed triangle?
• Cone side
  Calculate the volume and area of the cone whose height is 10 cm and the axial section of the cone has an angle of 30 degrees between height and the cone side.
• Triangle in a square
  In a square ABCD with side a = 6 cm, point E is the center of side AB, and point F is the center of side BC. Calculate the size of all angles of the triangle DEF and the lengths of its sides.
• EQL triangle
  Calculate the inradius and circumradius of an equilateral triangle with side a=77 cm.
• Calculate
  Calculate the area of triangle ABC, if given by alpha = 49°, beta = 31°, and the height on the c side is 9cm.
• Pentagon
  Calculate the length of side, circumference and area of a regular pentagon, which is inscribed in a circle with radius r = 6 cm.
• Right triangle
  Legs of the right triangle are in the ratio a:b = 2:8. The hypotenuse has a length of 87 cm. Calculate the perimeter and area of the triangle.
• Ratio of sides
  The triangle has a circumference of 21 cm, and the length of its sides is in a ratio of 6: 5: 3. Find the length of the longest side of the triangle in cm.
• Glass
  How many glass are needed to produce glass with base regular 5-gon if one base triangle in the base is 4.2 square cm and the height is 10 cm?
• Pyramid 8
  Calculate the volume and the surface area of a regular quadrangular pyramid with the base side 9 cm and side wall with the base has an angle 75°.
• Calculate
  Calculate the length of a side of the equilateral triangle with an area of 50cm².
• Inscribed and described circle
  Find the radii of a circle inscribed and circumscribed by a regular pentagon whose side measures 3 cm.
• Equilateral triangle vs circle
  Find the area of an equilateral triangle inscribed in a circle of radius r = 9 cm. What percentage of the circle area does it occupy?
• Side lengths
  In the triangle ABC, the height to the side a is 6cm. The height to side b is equal to 9 cm. Side "a" is 4 cm longer than side "b". Calculate the side lengths a, b.
• Triangle ABC v2
  Area of the triangle is 12 cm square. Angle ACB = 30º , AC = (x + 2) cm, BC = x cm. Calculate the value of x.
• Centre of the hypotenuse
  For the interior angles of the triangle ABC, alpha beta and gamma are in a ratio of 1: 2: 3. The longest side of the AB triangle is 30 cm long. Calculate the perimeter of the triangle CBS if S is the center of the side AB.
• ABCD square
  In the ABCD square, the X point lies on the diagonal AC. The length of the XC is three times the length of the AX segment. Point S is the center of the AB side. The length of the AB side is 1 cm. What is the length of the XS segment?