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How to find the area of a triangle, rectangle, circle, and other shapes. All area formulas

02 02 2024

02 02 2024

All area formulas

How to find the area of a triangle, rectangle, circle, and other shapes

Calculating the area of a figure is a fundamental concept in school geometry. Each specific shape has its own distinct formula. Mathema provides instructions on how to determine the area of a triangle, parallelogram, rhombus, circle, and various other geometric shapes. Keep this article for reference and utilize it as a handy guide for area formulas required in the mathematics section of your standardized exams.

Area of a Triangle

How to calculate the area of a triangle?

To find the area of a triangle, you can use the formula: it’s half of the base multiplied by the height drawn to that base. Alternatively, you can use Heron’s formula, which involves finding half of the triangle’s perimeter.


\[S=\frac12ah\] \[S=\sqrt{p(p-a)(p-b)(p-c)}\]
  • a — represents one of the sides of a triangle
  • h — stands for the height of a triangle
  • p — denotes half of the triangle’s perimeter

How to find the area of an equilateral triangle?

In an equilateral triangle, all angles measure 60°, and all sides have the same length. You can calculate the area of an equilateral triangle using various formulas:


\[S=\frac{a^2\sqrt3}4\] \[S=\frac12ah\]
  • a — one of the triangle’s sides
  • h — the triangle’s height

How to find the area of an isosceles triangle?

In an isosceles triangle, two sides have equal lengths, and the third side is referred to as the base. The area of an isosceles triangle is half of the base multiplied by the height.


\[S=\frac12ah\]
  • a — represents the base of a triangle
  • h — signifies the height of a triangle

How to find the area of a right triangle?

In a right triangle, one of the angles measures 90°. The sides that create this angle are referred to as the legs, and the opposite side is called the hypotenuse. The area of a right triangle is half of the product of its legs.


\[S=\frac12ab\]
  • a — the base of a triangle
  • h — the height of a triangle

How to find the area of a right triangle?

In a right triangle, one of the angles is a 90° angle. The sides that create this angle are referred to as the legs, while the opposite side is known as the hypotenuse. To calculate the area of a right triangle, you can use the formula: it’s half of the product of the lengths of its legs.


\[S=\frac12ab\]
  • a, b — the legs

Area of a Quadrilateral

How to find the area of a square?

A square is a four-sided shape with all sides equal in length, and all angles measuring 90°. The formula to determine the area of a square is perhaps the simplest among all geometry formulas; it is equal to the square of its side length.


\[S=a^2\]
  • a — length of one side of the square

How to find the area of a rectangle?

A rectangle is a four-sided shape where adjacent sides form right angles. While a square shares these characteristics, a rectangle is specifically identified by having unequal adjacent sides. The area of a rectangle is calculated by multiplying the lengths of its adjacent sides.


\[S=ab\]
  • a, b — lengths of the adjacent sides of the rectangle

How to find the area of a parallelogram?

A parallelogram is a four-sided shape with opposite sides that are parallel and equal in length. Opposite angles are also equal. The area of a parallelogram is found by multiplying one of its sides by the height drawn to that side. Alternatively, it can be calculated as the product of two adjacent sides, multiplied by the sine of the angle between them.


\[S=ah\] \[S=ab\sin\gamma\]
  • a, b — lengths of adjacent sides of the parallelogram
  • γ — angle between the adjacent sides of the parallelogram
  • h — height of the parallelogram

How to find the area of a rhombus?

A rhombus is a parallelogram with all sides equal in length. Several methods exist to determine the area of a rhombus. One approach is to find half of the product of its diagonals. Another method involves calculating the area as the product of a side and the height drawn to that side.


\[S=\frac12d_1d_2\] \[S=ah\]
  • d — lengths of the diagonals of the rhombus
  • a — length of a side of the rhombus
  • h — height of the rhombus

How to find the area of a trapezoid?

A trapezoid is a four-sided shape with two parallel sides and two non-parallel sides. The parallel sides are referred to as the bases of the trapezoid. To find the area of a trapezoid, including an isosceles trapezoid, you take half of the sum of its bases and multiply it by the height.


\[S=\frac{a+b}2h\]
  • a, b — lengths of the bases of the trapezoid
  • h — height of the trapezoid

How to find the area of a circle?

To calculate the area of a circle, you square its radius and multiply it by the mathematical constant π (pi).


\[S=\pi R^2\]
  • R — radius of the circle
  • π — Pi, approximately 3.14

How to find the surface area of a sphere?

To determine the surface area of a sphere, you square its radius, multiply it by the mathematical constant π (pi), and then multiply the result by four.


\[S=4\pi R^2\]
  • R — radius of the sphere
  • π — Pi, approximately 3.14

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How to find the area of a triangle, rectangle, circle, and other shapes. All area formulas

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Calculating the area of a figure is a fundamental concept in school geometry. Each specific shape has its own distinct formula. Mathema provides instructions on how to determine the area of a triangle, parallelogram, rhombus, circle, and various other geometric shapes. Keep this article for reference and utilize it as a handy guide for area formulas required in the mathematics section of your standardized exams.

Area of a Triangle

How to calculate the area of a triangle?

To find the area of a triangle, you can use the formula: it’s half of the base multiplied by the height drawn to that base. Alternatively, you can use Heron’s formula, which involves finding half of the triangle’s perimeter.


\[S=\frac12ah\] \[S=\sqrt{p(p-a)(p-b)(p-c)}\]
  • a — represents one of the sides of a triangle
  • h — stands for the height of a triangle
  • p — denotes half of the triangle’s perimeter

How to find the area of an equilateral triangle?

In an equilateral triangle, all angles measure 60°, and all sides have the same length. You can calculate the area of an equilateral triangle using various formulas:


\[S=\frac{a^2\sqrt3}4\] \[S=\frac12ah\]
  • a — one of the triangle’s sides
  • h — the triangle’s height

How to find the area of an isosceles triangle?

In an isosceles triangle, two sides have equal lengths, and the third side is referred to as the base. The area of an isosceles triangle is half of the base multiplied by the height.


\[S=\frac12ah\]
  • a — represents the base of a triangle
  • h — signifies the height of a triangle

How to find the area of a right triangle?

In a right triangle, one of the angles measures 90°. The sides that create this angle are referred to as the legs, and the opposite side is called the hypotenuse. The area of a right triangle is half of the product of its legs.


\[S=\frac12ab\]
  • a — the base of a triangle
  • h — the height of a triangle

How to find the area of a right triangle?

In a right triangle, one of the angles is a 90° angle. The sides that create this angle are referred to as the legs, while the opposite side is known as the hypotenuse. To calculate the area of a right triangle, you can use the formula: it’s half of the product of the lengths of its legs.


\[S=\frac12ab\]
  • a, b — the legs

Area of a Quadrilateral

How to find the area of a square?

A square is a four-sided shape with all sides equal in length, and all angles measuring 90°. The formula to determine the area of a square is perhaps the simplest among all geometry formulas; it is equal to the square of its side length.


\[S=a^2\]
  • a — length of one side of the square

How to find the area of a rectangle?

A rectangle is a four-sided shape where adjacent sides form right angles. While a square shares these characteristics, a rectangle is specifically identified by having unequal adjacent sides. The area of a rectangle is calculated by multiplying the lengths of its adjacent sides.


\[S=ab\]
  • a, b — lengths of the adjacent sides of the rectangle

How to find the area of a parallelogram?

A parallelogram is a four-sided shape with opposite sides that are parallel and equal in length. Opposite angles are also equal. The area of a parallelogram is found by multiplying one of its sides by the height drawn to that side. Alternatively, it can be calculated as the product of two adjacent sides, multiplied by the sine of the angle between them.


\[S=ah\] \[S=ab\sin\gamma\]
  • a, b — lengths of adjacent sides of the parallelogram
  • γ — angle between the adjacent sides of the parallelogram
  • h — height of the parallelogram

How to find the area of a rhombus?

A rhombus is a parallelogram with all sides equal in length. Several methods exist to determine the area of a rhombus. One approach is to find half of the product of its diagonals. Another method involves calculating the area as the product of a side and the height drawn to that side.


\[S=\frac12d_1d_2\] \[S=ah\]
  • d — lengths of the diagonals of the rhombus
  • a — length of a side of the rhombus
  • h — height of the rhombus

How to find the area of a trapezoid?

A trapezoid is a four-sided shape with two parallel sides and two non-parallel sides. The parallel sides are referred to as the bases of the trapezoid. To find the area of a trapezoid, including an isosceles trapezoid, you take half of the sum of its bases and multiply it by the height.


\[S=\frac{a+b}2h\]
  • a, b — lengths of the bases of the trapezoid
  • h — height of the trapezoid

How to find the area of a circle?

To calculate the area of a circle, you square its radius and multiply it by the mathematical constant π (pi).


\[S=\pi R^2\]
  • R — radius of the circle
  • π — Pi, approximately 3.14

How to find the surface area of a sphere?

To determine the surface area of a sphere, you square its radius, multiply it by the mathematical constant π (pi), and then multiply the result by four.


\[S=4\pi R^2\]
  • R — radius of the sphere
  • π — Pi, approximately 3.14

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